Number relationships form the final strand of number sense. This strand looks at numbers and quantities in connection with others. “Students understanding of quantity is based on their understanding of number meaning and number relationships,” (Chapin & Johnson, 2006) number relationships allow students to get the big picture of the quantities they are working with and understand how those quantities interact. Exploring number relationships builds on students’ ability to count and subitize, use place value and understand the different types of numbers to use numbers effectively. Once students are confident counting and with the number sequence they can begin to order and compare numbers. Students who have a good understanding of place value can begin to compose and decompose numbers first using the place value chart then extending their understanding to compose numbers in a wider variety of ways. Students also need to build their understanding of part / whole relationships seeing how quantities add together to create a whole. Factoring is also part of studying number relationships and extends students understanding of composition and decomposition from additive to multiplicative building skills for multiplication and division. Number relationships also encompass ratios and percentages, both of which compare quantities. Ratios compare two quantities to each other while percentages show parts of a whole.
Composition and Decomposition
Students require an understanding of the following skills to begin composing and decomposing numbers:
- Composition requires the knowledge that:
- numbers are made up of parts.
- two quantities can be combined to make a larger number.
- pairs and groups that make 5, 10 and 20.
- place value.
- base ten numeration.
- Decomposition is the inverse of composition, with students understanding that:
- Quantities can be split into smaller parts.
- pairs and groups that make 5, 10 and 20.
- place value.
- Base ten numeration.
Part / whole relationships
Factors
Factoring is another form of part whole relationships, where the common part-part-whole structure takes an additive approach to add the parts together to get the total factoring uses a multiplicative approach. When looking for factors of a number, students are trying to determine what numbers multiply together to create the total quantity. Students can discover factors by building or drawing arrays and looking what the possible arrays are for each number. Going through this process students will discover that some numbers only build one array, these are the prime numbers. Prime numbers are the numbers that can only be factored by themselves and one, common primes include 2,3,5,7,11 and 13. Noting that except for 2 all of the primes are odd numbers.
Comparing
Comparing numbers is so much more than the “alligator mouths” comparison, working with inequalities is a study in number relationships that solidifies student number sense. It is not enough in many situations to know how many are in a group, it also becomes important to know which group is bigger or smaller. Think of purchasing classroom supplies, if you have 30 students and 25 pencils you don’t have enough pencils, you did not determine this using subtraction you quickly compared the two numbers mentally and recognized that you have more students than pencils. Comparison is an essential life skill and becomes a tool for estimation as students establish their ability to compare numbers and determine what is too big or too small.
Ordering
“Perhaps one of the most interesting “technologies” for keeping track of large numerosity’s, and certainly one of the most powerful tools for teaching about them, is the number line,” (Krasa, Tzanetopoulos, & Maas, 2023) number lines are a visual tool that allows students to build the relational piece of their number sense. Ordering numbers builds on students’ understanding of the counting sequence but cannot rely solely on it. Students need to develop the ability to order numbers from least to greatest with. Ordering extends students’ ability to compare numbers. Once students can compare two numbers, they can begin to use those skills to put multiple numbers in order.
Ratios
Ratios are a way of representing the relationship between two quantities and can represent a part-to-part relationship or a part to whole relationship. Students will encounter ratios in many real-life situations from the odds of winning a specific prize in a draw to cooking or working in many trades. Gearing in engines or on bikes relies on ratios, many food items are prepared according to ratios, concrete and household chemicals are mixed using ratios.
- Part to Part Ratios
- Connects two or more parts
- Common relationships
- Cooking
- Geometry
- Graphing
- Algebra
- Probability
- Compares two or more related parts
- Can be part of composition and decomposition
- Part to whole ratios
- Compares parts to a whole
- Percentages are a specific type of part to whole ratio
- May be written as a fraction
- Ratios as quotients
- Relates to the number of objects you can get for a specified amount of something else
- Can be divided to get the unit rate
- Ratios as rates
- Describes how much usage can be gotten from a single unit
- Examples
- Liters of gas per 100km
- Square feet of wall per gallon of paint
- Eggs per carton
- Currency conversions
- Interest
- Taxes
- Relates measurements to each other
- Centimeters to meters
- Inches to meters
- Represents an infinite set of equivalent ratios
Percentages
A percentage is a specific type of ratio, relating to a total of 100 units. Percentages can also be expressed as decimal hundredths or fractions with a denominator of 100. Percentages are helpful for communicating in a standard form, as they all reference the same denominator it is easier to make a comparison. Traditionally grades have been represented as percentages making it easy to monitor a student’s progress and compare students’ scores when needed. Percentages can also be used to communicate rates such as sales.