Pariksha SaathiLesson 7 of 8 · Mathematics, Classes I-V (Paper I)
Patterns and Data Handling: Number and Shape Patterns, Tally Marks, Pictographs and Bar Graphs
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Why patterns matter
Seeing, extending and describing patterns is the beginning of algebraic thinking, and Math-Magic returns to it in every class, for example in Play with Patterns in Classes 3 and 4 and Can You See the Pattern? in Class 5. Children first copy a pattern, then extend it, then make their own and finally put its rule into words. In a repeating pattern the smallest part that repeats is the core: red, blue, blue, red, blue, blue has the core red, blue, blue of length 3, so the 20th bead is found from 20 = 3 × 6 + 2; it is the 2nd item of the core, blue. Shape patterns include borders, rangoli, block prints, tilings and shapes that turn by a fixed amount at each step, such as an arrow turning a quarter turn clockwise each time, which points the same way again at every 4th step.
Growing number patterns
In 3, 7, 11, 15 ... the rule is add 4, so the next term is 19 and the 10th term is 3 + 9 × 4 = 39. When the differences change, look at the differences themselves: 2, 6, 12, 20, 30 has differences 4, 6, 8, 10, so the next term is 42 (the terms are 1 × 2, 2 × 3, 3 × 4 and so on). Square numbers 1, 4, 9, 16, 25 are dots arranged in squares; triangular numbers 1, 3, 6, 10, 15 are dots in triangles, and two consecutive triangular numbers make a square number (6 + 10 = 16). Doubling gives 1, 2, 4, 8, 16, 32. Patterns in calculation invite children to predict before calculating: 37 × 3 = 111, 37 × 6 = 222 and 37 × 9 = 333, and 11 × 11 = 121 while 111 × 111 = 12,321.
Patterns on grids and calendars
On a 1 to 100 number grid, a step to the right adds 1, a step down adds 10 and a step diagonally down to the right adds 11. On a calendar page a step down adds 7, so in any 2 by 2 block of dates the two diagonals have equal sums, and in any 3 by 3 block the 9 dates add up to 9 times the middle date (with 15 in the middle the block adds up to 135). Questions of this kind test whether the teacher can see why a pattern works, not only that it works.
Collecting and organising data
Data handling follows a cycle: ask a question that matters to the children (How do we come to school? In which month are our birthdays?), collect the data by counting, a survey or observation, organise it in a list or table, show it as a picture or graph, and then discuss what it tells. The Smart Charts chapters of Classes 3, 4 and 5 work in this way. Tally marks make counting easier: each item gets one stroke and every fifth stroke is drawn across the four before it, so the strokes form bundles of 5 that can be counted 5, 10, 15; a count of 23 shows 4 bundles and 3 single strokes.
Pictographs
A pictograph uses a picture or symbol for each item or group of items, and its key (scale) says what one symbol stands for. If one book symbol stands for 10 books borrowed, 3 symbols on Monday mean 30 books, 5 and a half symbols on Tuesday mean 55 and 2 symbols on Wednesday mean 20; the total is 105, and Tuesday exceeds Wednesday by 35. The key should suit the numbers: values of 40, 25 and 35 can be drawn with 1 symbol for 5, giving 8, 5 and 7 symbols. Counting the symbols without reading the key is the commonest wrong answer.
Bar graphs
A bar graph shows each category as a bar of equal width with equal gaps between bars; the length of a bar shows the value against a scale on the axis, and the axes, the scale and the title must be labelled. The favourite fruits of a class (mango 18, banana 12, apple 9, guava 6) drawn with 1 cm for 3 children give bars of 6, 4, 3 and 2 cm. From the graph children can read that 45 children were asked, that mango is the most popular, that twice as many chose mango as apple and that 2/5 of the class chose mango. Good questions move from reading one value to comparing (how many more?), combining (how many in all?) and deciding (which fruit should the school buy most of?).
Key facts
- Repeating pattern red, blue, blue: the core has length 3, and the 20th bead is blue because 20 = 3 × 6 + 2.
- 3, 7, 11, 15 ... adds 4 each time: the next term is 19 and the 10th term is 3 + 9 × 4 = 39.
- 2, 6, 12, 20, 30, 42: differences 4, 6, 8, 10, 12; the terms are 1 × 2, 2 × 3, 3 × 4 and so on.
- Square numbers 1, 4, 9, 16, 25; triangular numbers 1, 3, 6, 10, 15; two consecutive triangular numbers make a square: 6 + 10 = 16.
- Calculation patterns: 37 × 3 = 111, 37 × 6 = 222, 37 × 9 = 333; 111 × 111 = 12,321.
- Calendar: in a 3 by 3 block of dates the sum is 9 times the middle date (middle 15 gives 135).
- Tally marks come in bundles of 5 (four strokes and a fifth across them); 23 is 4 bundles and 3 strokes.
- Pictograph with key 1 symbol = 10 books: 5 and a half symbols mean 55 books; always read the key first.
- Bar graph of mango 18, banana 12, apple 9, guava 6 at 1 cm = 3 children: bars of 6, 4, 3 and 2 cm; 45 children in all.
- Data cycle: ask a question, collect, organise (tally, table), represent (pictograph, bar graph) and interpret.
Quick revision
Patterns: copy, extend, create, describe the rule; a repeating pattern has a core; 3, 7, 11 ... has 10th term 39; squares 1, 4, 9, 16; triangular numbers 1, 3, 6, 10; number grid: right +1, down +10; calendar: down +7, a 3 by 3 block = 9 × middle date; data: question, collect, tally in 5s, pictograph with a key, bar graph with equal widths and a scale; read, compare, total, decide.