What are Patterns in Numbers?
A set of colors, numbers, shapes, behaviors, or other things that generally repeats itself is referred to as a pattern. They might be finite or infinite, and they can be tied to any event or object. A pattern in numbers is a repeated set of numbers based on any mathematical operation. One of the examples of a number pattern is repeatedly adding 10.
What are Place Values and why is the need for Place Values?
Every digit in a number has a place value. The value indicated by a digit in a number based on its position in the number is known as the place value.
Here’s an example that shows the relationship between the place or position, and the place value of the digits in a number. The value of each digit in a number is called the place value. Understanding place value helps us in calculating a number’s value.
100s, 10s, and 1s in the place value system:
The entire number system is based on place value. This is a system in which the value of a number is determined by the position of its digits. One of the next larger units after the one’s place is tens and is made up of ten ones.
One of the next larger units, hundreds, is made up of ten of the previous units, which is tens. This pattern holds true for larger numbers, as we go from hundreds to thousands, to ten thousands, and even in larger numbers from there on.
For example,
ten ones = one ten
ten tens = one hundred
ten hundreds = one thousand,
ten thousands = one ten thousand
It is this pattern that keeps repeating itself.
A number can be expressed as a sum of its place values in the following way. Let’s take the number 578. This is how we express the number as a sum of its place values.
578 = 500 + 70 + 8
Now, let us try to visualize place value using some models.
Consider a number, say 578, this is what the value of each digit would look like. Each digit’s individual value is lesser than the previous one as you move from left to right. But, with respect to their position, the value of each digit is as follows.
To understand place value patterns, look at how each place value has a cube pattern, the ones place has individual cubes. The tens place has rods of tens cubes according to the value in the ten’s place. In the hundreds place, rods are arranged in tens giving us sheets made of single cubes. The number of the repeated pattern of sheets depends on the digit at the hundreds place.
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Numerical Problems involving various Place Value patterns
Determine the next number based on the pattern of increase in the place values in the following sets of numbers.
Example 1:
123, 124, 125, ___,___,___
Solution:
The ones place is increasing by one number in each of the digits in the sequence. So the next three consecutive numbers which follow this pattern are: 126, 127, 128.
Example 2:
111, 211, ___, 411, 511, ___, ___
Solution:
In the pattern of numbers shown here, the hundreds place increases by one. So, the complete pattern of numbers is as follows.
111, 211, 311, 411, 511, 611, 711.
Example 3:
Write the next three numbers corresponding to the pattern of the blocks shown below.
Solution:
The numbers represented here are 311, 321, and 331. As each number progresses, the tens place increases by 1. So, the next three numbers would be 341, 351, and 361.
Example 4:
Represent the following number as a sum of their place values: 145
Solution:
One is in the hundreds place, four in the tens place, and 5 in the ones place so, the sum of the number would look like: 100 + 40 + 5 = 145.
Example 5:
Represent 646 as the sum of the individual numbers according to place value.
Solution:
Six is in the hundreds place, four is in the tens place, and 2 is in ones place. The sum of these numbers would look like, 600 + 40 + 6 = 646
Real-life Modeling Problem:
Example 6:
There are 100 students assembled at the school canteen. Three more classes joined the remaining students and now there are a total of 130 students. Determine the number of students in each class, if the number of students is equal.
Solution:
Initially, there were a hundred students in the canteen. Three more classes joined in and this increased the number of students from 100 to 130. When three classes joined, 30 students were increased.
So each class holds 10 students.
Example 7:
In a competition, each participant earns points according to the number of apples thrown into baskets that have a prescribed value.
Jenson throws three apples in the blue basket in three attempts, four apples in the yellow basket in four attempts, and five apples in the red basket in five attempts. The score he gets is 345 in 12 throws. How many points is each basket worth?
Solution:
Jenson scored a total of 345 points. The number of attempts he gets is equal to the digits in each place of the number 345. After expressing the number in place values, we get the following result.
345 = 300 + 40 + 5
Jenson threw three apples into the blue basket, this means that the blue basket is worth 300/3 = 100 points.
Jenson threw four apples into the yellow basket, which means that the blue basket is worth 40/4 = 10 points
Jenson threw five apples into the red basket, which means that the blue basket is worth 5/5 = 5 points
Example 8:
Joseph has a collection of multi- coloured crazy balls in a box. He had 743 balls in the box. He added 60 balls into the box. Represent the existing set of balls in the form of base ten blocks and also show how many sets of ten balls were added to the collection.
Solution:
There were 743 balls initially. First, we will express the number in its place value form.
743 = 700 + 40 + 3
So, 7 fall under the hundreds place, 4, under tens place and three under ones place.
Sixty balls were added to the collection. Six sets of tens make sixty. Adding six blocks to the tens place would look like this.
Now, there are a total of ten blocks in the units place, hence one gets carried over to the hundreds place. In other words, a hundred blocks will be added to the hundreds place.
The final figure would look like the following.
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The place value of 0 in any number is always taken as 0.
For example, the place value of 0 in 100 is zero and the place value of 0 in 707 is also zero.
Learning the concept of place value helps us in:
- Addition
- Subtraction
- Multiplication
- Division