This is an open-ended question. A valid answer should name everyday situations where counting, measuring, comparing, estimating or calculating is used.
Yes. Mathematics helps in shopping, cooking, reading time, measuring distance, counting money, planning travel, sharing food equally, measuring cloth, designing buildings and comparing scores in games.
The chapter explains that finding mathematical patterns and their explanations helps people build theories and applications. Examples include gravitation for space travel and genome patterns for diagnosing and curing diseases.
Mathematics helps humanity by making measurement, prediction, design and planning possible. It is used in scientific experiments, engineering bridges and buildings, designing machines, creating calendars and clocks, running banks and economies, analysing election data, making computers and mobile phones, and planning transport systems such as trains, cars and planes.
Read each sequence by comparing consecutive terms. Some grow by adding a fixed or changing amount, while powers grow by repeated multiplication.
Yes. All 1's: every term is 1. Counting numbers: add 1 each time. Odd numbers: add 2 starting from 1. Even numbers: add 2 starting from 2. Triangular numbers: add the next counting number. Squares: $1^2,2^2,3^2,\ldots$. Cubes: $1^3,2^3,3^3,\ldots$. Virahanka numbers: each term after the first two is the sum of the previous two. Powers of 2: multiply by 2 each time. Powers of 3: multiply by 3 each time.
Apply the rule for each sequence. For example, triangular numbers continue by adding 8, 9 and 10 to 28; powers of 2 continue by multiplying by 2; powers of 3 continue by multiplying by 3.
All 1's: next three are 1, 1, 1. Counting numbers: 8, 9, 10. Odd numbers: 15, 17, 19. Even numbers: 16, 18, 20. Triangular numbers: 36, 45, 55. Squares: 64, 81, 100. Cubes: 343, 512, 729. Virahanka numbers: 34, 55, 89. Powers of 2: 128, 256, 512. Powers of 3: 2187, 6561, 19683.
Table 2 shows the first five pictures for each sequence. The next picture is the sixth picture in that sequence, so use the sixth term of each corresponding number sequence.
The next pictures should show: squares with 36 dots arranged as $6 \times 6$; triangular numbers with 21 dots in 6 rows; cubes with 216 unit cubes as a $6 \times 6 \times 6$ cube; one more single dot for All 1's; 6 dots for counting numbers; 11 dots for odd numbers; and 12 dots for even numbers.
The names come from the shapes formed by arranging dots or unit cubes.
The numbers 1, 3, 6, 10, 15, ... are called triangular numbers because that many dots can be arranged as triangles. The numbers 1, 4, 9, 16, 25, ... are called square numbers because they can be arranged as square arrays: $1 \times 1,2 \times 2,3 \times 3,\ldots$. The numbers 1, 8, 27, 64, 125, ... are called cubes because they can be arranged as solid cubes: $1^3,2^3,3^3,4^3,5^3,\ldots$.
The square arrangement has $6^2=36$ dots. The triangular arrangement has $1+2+3+4+5+6+7+8=36$ dots.
Draw 36 dots once as a $6 \times 6$ square and once as a triangle with 8 dots in the bottom row, then 7, 6, 5, 4, 3, 2 and 1 dots above it.
The differences are $7-1=6$, $19-7=12$, and $37-19=18$. The differences increase by 6, so the next difference is 24. Therefore, the next number is $37+24=61$.
They are hexagonal numbers. The next number is $61$.
The key idea is repeated multiplication: powers of 2 double each time and powers of 3 triple each time.
Powers of 2 can be shown by repeatedly doubling a picture: 1 dot, 2 dots, 4 dots, 8 dots, 16 dots, and so on. Powers of 3 can be shown by repeatedly tripling: 1 dot, 3 dots, 9 dots as a $3 \times 3$ square, 27 dots or cubes, and so on.
The sum of the first $n$ odd numbers is $n^2$. For $n=10$, the sum is $10^2=100$.
The sum of the first 10 odd numbers is $100$.
The sum of the first $n$ odd numbers is $n^2$. For $n=100$, the sum is $100^2=10000$.
The sum of the first 100 odd numbers is $10000$.
For a square of side $n$, count dots row by row as $1+2+\cdots+n+\cdots+2+1$. This total is $n^2$, so the result is a square number.
Yes. Arrange dots in a square and divide them into rows that increase up to the middle row and then decrease. For example, $1+2+3+2+1=9$ makes a $3 \times 3$ square and $1+2+3+4+3+2+1=16$ makes a $4 \times 4$ square.
The expression counts the rows of a $100 \times 100$ square: $1+2+\cdots+99+100+99+\cdots+2+1=100^2=10000$.
The value is $10000$.
The running sums are $1$, $1+1=2$, $1+1+1=3$, and so on. The up-and-down sums are $1$, $1+1+1=3$, $1+1+1+1+1=5$, and so on.
Adding the All 1's sequence up gives the counting numbers: $1,2,3,4,5,\ldots$. Adding the All 1's sequence up and down gives odd numbers: $1,3,5,7,9,\ldots$.
The sums are $1$, $1+2=3$, $1+2+3=6$, $1+2+3+4=10$, and so on. These can be drawn as triangles with rows of 1, then 2, then 3 dots, and so on.
Adding the counting numbers gives the triangular numbers: $1,3,6,10,15,\ldots$.
The sums are $1+3=4$, $3+6=9$, $6+10=16$, $10+15=25$. Two consecutive triangular dot patterns can be fitted together to form a square.
You get square numbers: $4,9,16,25,\ldots$.
Each partial sum of powers of 2 is one less than the next power of 2: $1=2-1$, $1+2=4-1$, $1+2+4=8-1$, and so on.
The sums are $1,3,7,15,31,\ldots$. After adding 1 to each, we get $2,4,8,16,32,\ldots$, the powers of 2.
Using triangular numbers $1,3,6,10,15,\ldots$, compute $6T+1$: $6(1)+1=7$, $6(3)+1=19$, $6(6)+1=37$, $6(10)+1=61$, $6(15)+1=91$. These match the hexagonal-number pattern.
You get $7,19,37,61,91,\ldots$, which are hexagonal numbers after the first hexagonal number 1.
The sums are $1=1^3$, $1+7=8=2^3$, $1+7+19=27=3^3$, and $1+7+19+37=64=4^3$. A cube can be built by adding hexagonal layers around the previous cube.
You get cube numbers: $1,8,27,64,\ldots$.
The first pattern can be shown by adding L-shaped layers around a square. The second pattern can be shown by pairing dots in each counting-number group.
One pattern is that every square number is the sum of consecutive odd numbers. For example, $16=1+3+5+7$ and $25=1+3+5+7+9$. Another pattern is that even numbers are twice the counting numbers: $2,4,6,8,\ldots=2\times1,2\times2,2\times3,2\times4,\ldots$.
The shape sequences are formed by repeating a rule, just as number sequences are formed by repeating a numerical rule.
Yes. Regular polygons gain one side each time. Complete graphs add one point each time and connect every pair of points. Stacked triangles grow by adding a larger triangular layer. Stacked squares grow by adding a larger square layer. Koch snowflakes replace every line segment by a smaller speed-bump pattern at each step.
Each sequence has a repeatable rule. Some next shapes are easy to draw exactly, while shapes such as the Koch snowflake become difficult because the segments get very small.
The next regular polygon is an 11-sided regular polygon. The next complete graph is $K_7$. The next stacked triangle and stacked square add the next larger layer. The next Koch snowflake is made by replacing every line segment of the previous shape with the same smaller speed-bump pattern. These are possible to draw, though the Koch snowflake becomes increasingly detailed.
A polygon has one corner where each pair of adjacent sides meets. Therefore, in every polygon, the number of sides equals the number of corners.
The number of sides is $3,4,5,6,7,8,9,10,\ldots$. The number of corners is also $3,4,5,6,7,8,9,10,\ldots$.
In a complete graph, every pair of points is joined by a line. As points are added, the number of new lines added is $1,2,3,4,\ldots$, so the totals are triangular numbers.
The number of lines is $1,3,6,10,15,\ldots$, the triangular-number sequence.
Each stacked square is an $n \times n$ arrangement of little squares, so the number of little squares is $n^2$.
The numbers are $1,4,9,16,25,\ldots$, the square-number sequence.
The rows in the stacked triangles can be counted as $1$, then $1+2+1=4$, then $1+2+3+2+1=9$, and so on. These up-and-down sums give square numbers.
The numbers are $1,4,9,16,25,\ldots$, the square-number sequence.
The first shape has 3 line segments. At each step, every line segment is replaced by 4 smaller line segments, so the number is multiplied by 4 each time. Thus the sequence is $3,3\times4,3\times4^2,3\times4^3,\ldots$.
The total numbers of line segments are $3,12,48,192,768,\ldots$.