Children discover relationships between numbers, beginning with their first experiences of counting their fingers. This finger counting helps them to see the relationship of the numbers 1–4 to 5, and understanding that relationship (of 1–4 to 5) helps them to recognize that a five frame with counters in all but one square represents the number 4. This understanding of the relationship of 1–4 to 5 is later extended to an understanding of the relationship between all of the numbers from 0 to 10 and, particularly, the relationship of the numbers from 0 to 10 with the anchors of 5 and 10. This understanding is an important founda- tion for an understanding of the larger numbers. If students have worked with combinations of 10, they have a better understanding of problems that involve operations with larger numbers. For example, in adding 27 and 35, if they know that 7 is 3 away from 10, they can take the 3 from 35, add it to 27 to make 30, add on the 30, and then add the 2 that remains. This understanding of the 10 relationships can be extended to an understanding of 100’s, and the patterns for 1 to 9 provide a useful guidepost for learning how to count by 100’s and then by 1000’s.
Understanding the relationships of other numbers to these anchor numbers of 5 and 10 helps students later to understand place value (see also “Representations” for a discussion of place value). The relation- ships between 1’s and 10’s, 10’s and 100’s, 100’s and 1000’s, and so on, are important in understanding number and the operations on number. Using base ten materials or other proportional manipulatives helps students see the relationships between these quantities. The base ten materials consist of single units (ones), rods (tens), flats (hundreds), and cubes (thousands). Ten of the ones units are put together to form 1 tens rod, 10 rods are put together to form 1 hundreds flat, and 10 flats
of hundreds form a thousands cube. Knowing the relationships in place value (as the digits move to the left, numbers go up by a factor of 10; as digits move to the right, numbers decrease by a factor of 10) helps students work efficiently and effectively with numbers. It is important to note that, although proportional manipulatives help students develop their concept of place value, the concept is not inherent in the manipulatives. Students must develop the concept through interaction with the materials, using the materials as tools for building up the concept. Showing students models of a concept – for example, two-digit addition – using base ten materials without allowing the students to develop the concept of two-digit addition themselves is as ineffective as making them memorize a rote procedure without any understanding.
Characteristics of Student Learning and Instructional
Strategies by Grade
KINDERGARTEN
Characteristics of Student Learning
In general, students in Kindergarten:• learn in the beginning of Kindergarten to sort and classify objects according to attributes such as colour, size, and shape;
• gradually begin to estimate and count to see the relationships that help them to identify concepts of more, less, or the same as;
• develop a beginning understanding of: – 1 more than – 1 less than – 2 more than – 2 less than – anchors of 5 and 10 – part-part-whole relationships
• develop an understanding of the patterns in the number system and use this knowledge to learn the names of the numbers in the 20’s;
• can begin to see patterns in a hundreds chart (or carpet) and on the number line.
Instructional Strategies
Students in Kindergarten benefit from the following instructional strategies: • providing multiple experiences of comparing sets of objects and observing
their relationships. For example, look at sets of objects and compare them with a set amount: are they more than, the same as, or less than the amount?;
• providing experiences with finger plays that help students to recognize the relationships between 5 and 10 and the rest of the numbers from 0 to 10; • providing multiple experiences of composing and decomposing amounts –
for example, take 5 apart and look at the relationships of its many compo- nents: 0 + 5, 1 + 4, 2 + 3, 3 + 2, 4 + 1, 5 + 0. It is valuable for students to observe that, as one component number goes down, the other goes up; • providing many opportunities to use five frames, ten frames, hundreds
charts, and number lines to build up an understanding of the relationships between numbers.
GRADE
1
Characteristics of Student Learning
In general, students in Grade 1:• see relationships between two numbers (e.g., that 7 is 2 more than 5, 1 less than 8, or 3 away from 10);
• consolidate their understanding of part-part-whole relationships, recognizing that numbers can be separated into component parts (e.g., 5 is 1 + 4 or 2 + 3, and so on);
• develop an understanding of the relationships between numerals and the grouping or bundling of objects into tens and ones;
• recognize some of the relationships that are inherent in number lines and the hundreds chart. For example, counting by 2’s involves skipping every second number, and such an action produces a consistent pattern along the number line and down the hundreds chart. Students see other patterns in the hun- dreds charts (e.g., odd and even numbers, counting by 5’s and 10’s);
• see the relationship between wholes and halves (a whole can be partitioned into two equal parts, and each part is one-half) as they grow to understand part-part-whole relationships;
• begin to use their understanding of the relationships between numbers as strategies for the basic facts. For example, knowing the doubles (e.g., 2 + 2, 3 + 3) makes learning the doubles plus one easier (e.g., 2 + 3 is double 2 plus 1 more).
Instructional Strategies
Students in Grade 1 benefit from the following instructional strategies:
• providing multiple experiences of composing and decomposing amounts. For example: decompose the teen and decade numbers into their many parts and look at the relationships of those parts (e.g., 20 is ten more than 10 and ten less than 30);
• providing many opportunities to use five frames, ten frames, hundreds charts (or carpets), number lines, and simple place-value charts to build up an understanding of relationships between numbers;
• building a hundreds chart for the purpose of exploring the patterns of the numbers and their relationships. Students can work together in groups or can work individually to build a chart.
GRADE
2
Characteristics of Student Learning
In general, students in Grade 2:• build on their ability to compare and order concrete materials in developing an understanding of larger numbers. For example, bundling tens and single units to understand two-digit numbers and then bundling hundreds, tens, and single units to understand three-digit numbers helps them to recognize the magnitude of the digits in the ones and tens positions;
• decompose larger numbers to get a sense of the relationships of a number to other single digits and to the decades (e.g., in adding 29 and 31, they may decompose 29 into 20 and 9 to make it easier to add to 31, and then add the 20 and the 30 together and the 1 and the 9 together, for the total of 60); • develop a sense of the relationships between the operations, recognizing that
addition is the inverse of subtraction;
• continue to use their understanding of the relationships between numbers to learn the basic and multidigit facts of addition and subtraction. For example, they may use the strategy of making tens to help with adding 18 and 6. They can add 18 to 2 of the 6 to make 20, and then add on the remaining 4. They can also use strategies such as compensation, which involves a good under- standing of the proportional relationship between sets of numbers. For exam- ple, knowing that the distance on the number line between 22 and 68 is the same as that between 20 (22 less 2) and 66 (68 less 2) allows them to turn a harder question into an easier one (especially when they are making the computation mentally);
• make comparisons between numbers and know from the larger digit in the tens place that a number like 92 is significantly larger than 29. This concep- tual understanding helps them begin to see relationships between larger numbers, between common fractions, and between the operations: addition, subtraction, multiplication, and division.
Instructional Strategies
Students in Grade 2 benefit from the following instructional strategies:
• providing experiences of composing and decomposing larger numbers, espe- cially numbers with a tens and/or hundreds digit (e.g., 56 can be understood as 5 tens and 6 ones or as 4 tens and 16 ones);
• providing experiences with bundling or grouping objects into 5’s (e.g., in tal- lies) and into 10’s (e.g., bundles of 10 craft sticks) to help students recognize the relationships between numbers, especially in understanding place value; • providing many opportunities to use ten frames, hundreds charts or carpets, number lines, arrays, and place-value charts to build up an understanding of relationships between numbers;
• working with hundreds charts to explore patterns in number. For example, adding by 10’s on a hundreds chart involves movement from one number to the one immediately below it; adding by 9’s involves movement from one number to the number immediately below and to the left of it; adding by 11’s involves movement from one number to the number immediately below and to the right of it.
GRADE
3
Characteristics of Student Learning
In general, students in Grade 3:• extend their understanding of the base ten number system to identify rela- tionships throughout that number system, from the decades to the 100’s to the 1000’s. This understanding of the relationships in number is still depend- ent upon concrete representations, especially for an understanding of place value;
• develop a sense of the relationships between the operations, recognizing that addition is the inverse of subtraction, multiplication is the inverse of division, and so on;
• use more abstract constructs – for example, a mental image of the hundreds chart or a number line – to help determine the relationship between such numbers as 51, 61, and 71;
• use manipulatives to develop their understanding of four-digit numbers, common fractions, and mixed numbers;
• develop a sense of the relationships between the operations, recognizing that addition is the inverse of subtraction, multiplication is the inverse of division, multiplication can be viewed as repeated addition, and division can be viewed as repeated subtraction;
• continue to use their understanding of the relationships between numbers to learn the basic and multidigit facts of addition, subtraction, multiplication, and division. For example, they may use their knowledge of the 5 times table to calculate the 6 times tables, or their knowledge of the 2 times tables to learn the 4 times tables.
Instructional Strategies
Students in Grade 3 benefit from the following instructional strategies:
• providing experiences of composing and decomposing larger numbers, espe- cially numbers with a tens or hundreds digit (e.g., 56 can be understood as 5 tens and 6 ones);
• providing experiences with bundling or grouping objects into 5’s (e.g., in tal- lies) and into 10’s (e.g., bundles of 10 craft sticks) to help students recognize the relationships between numbers, especially in understanding place value; • using ten frames, hundreds charts, number lines, arrays, and place-value
charts to build up an understanding of relationships between numbers; • working with hundreds chart to explore patterns in number. For example,
adding by 10’s on a hundreds chart involves movement from one number to the one immediately below it; adding by 9’s involves movement from one number to the number immediately below and to the left of it;
• encouraging students to use an “invisible” number line (or similar mental image) to help in developing more complex strategies for solving problems involving an understanding of the relationships between numbers. For exam- ple, in adding 50 and 22, they can move along a blank line, which represents the number line, to the decades, from 50 to 60 to 70, and then make 2 more single moves to come to 72.
Overview
A number is an abstract representation of a very complex concept. Numbers are represented by numerals, and a numeral can be used in many different ways. For instance, consider this sentence: “John, who is in Grade 1, is inviting 15 children to his 7th birthday on January 5, 2003, at 2 o’clock.” Think about the different ways that “number” is used in the statement.
The following are key points that can be made about representation in the primary years:
A numeral represents the number symbol, the number word, place- ment in a series of counts, placement on a number line, a place-value position, and a quantity of objects. The numeral 1, depending on its placement, can mean 1, 10, 100, 1000, and so on.
A very important aspect of understanding number is the connection between the symbol for a number or part of a number (e.g., a fraction or a decimal) and what that symbol represents with reference to quantity, position, or magnitude or size.
An important aspect of representation is learning how to read and write numerals and connect numerals with written and spoken words for numbers.