Grade 6 arithmetical science lesson plan
Kei Ohno

Research theme: Discuss a teaching method and materials on how to let any children find andcreate negative numbers through extracting necessity and inevitability of existence of thenegative numbers.
1. Lesson title: “Where should we place 0?” Introduction of negative numbers
2. The main point of this practice(1) Actual introduction and problems of positive and negative numbers at grade 7 (grade 1 at Japanese lower
secondary schools).
Generally in Japan, teachers use the negative temperature symbol “-” to introduce negative numbers. Thenchildren learn how to use the number line to express east-west transportation from a base point 0 with positive
and negative numbers. Through this challenge children have to understand that 0 does not mean an empty
space and each of them can place their 0s on their number lines. They also have to realise that the left side of
the 0 are negative numbers and the right side of the 0 are positive numbers.
It might be easy for the grade 7 students to learn negative numbers through above procedure. But
teachers give definitions and the symbol of negative numbers and they show the number line musthave the base point 0. I do not think this teaching and learning approach is suitable for children;
children’s curiosity, finding and creativity are neglected.
(2) Finding base point 0 to perceive the existence of negative numbers through necessity and inevitability
This lesson plan applies inflexibility of a primary arithmetic subtraction rule to extract learners’ curiosity. InJapanese primary schools, children learn positive rational numbers.
If we have two positive numbers: a, b and a > b. The following expressions are meaningful and children canmake calculations.
a + b, b + a (addition), a x b, b x a (multiplication), a ÷ b, b ÷ a (division)
But only subtraction is an exception. An expression a - b is meaningful and calculatable but a swapped
expression b - a is not calculatable for children under Japanese primary arithmetic curriculum because theanswer is negative number. If we deliberately let children encounter this inflexibility, they might think this iscurious and intentional. When you ask children the value difference between b and a, they can answer
instantly. We can let the children to express b - a using number line and “Where should we place 0?” becomes
necessary. If children can place 0 properly, they will be able to find the existence of numbers less than 0.
(3) Tools
A 2m tapeA 5 m tape(Please use a 20 cm long blue magnet tape and a 50 cm long red magnet tape.)
What is the length difference between two tapes?
 5 m - 2 m case 2m - 5m case00
3 m longer
3 m shorter
If we define 2m as 0, the 5 m tape is 3 m longer than 0. If we define 5m as 0, the 2 m tape is 3 m shorter than 0.
3. Objectives of this practical Through comparing to addition, division and multiplication, let children to find a subtraction expression which
has small value minuend than the subtrahend is not meaningful.
 Let the children understand the position of 0 on the number line and notice that the existence of numbers
which are less than 0.

1

4. Procedure of this practical
Flow of children’s learning
The teacher shows children a 20 cm long blue magnet tape written 2 m , and a 50cm long red magnet tape written 5 m
Teacher: Let us make some calculations using these two tapes.
Children: Connect the 2 m tape and the 5 m tape. 2 + 5= 7
Teacher: Express the answer (a 7 m tape) using the tapes.

027Children: We can swap the tapes. 5 + 2 = 7Children: The 5m tape is 2.5 times longer than the 2m tape. 5 ÷ 2 = 2.5

01

2.5

Children: We can swap them again. 2 ÷ 5 = 0.4

00.4

1

Children: A rectangle, 2 m wide and 5 m long, has an area of 10 m2.
2 m x 5 m = 10 m2
Children: We can swap them again. 5 m x 2 m = 10 m2
Children: The 5 m tape is 3 m longer than the 2 m tape. 5 - 2 =3Children: The length difference between 5 m and 2 m is 3 m.
Teacher: Where is 0 on the number line?
Children: If we define 2 m as 0, 5 m tape is 3.

0Remarks
 We can use magnet tapes
to visualise the questions
and explanations.
 Each time we use thenumber line, ask children“Where is the 0?”, “Where isthe answer “7?” to enhancechildren’s thinking.
 Keep on asking children tothink about the swappedexpressions. Let thechildren consider whether
the swapped expressions
are meaningful or not.
We cannot express theresult of multiplication using
the number line but letchildren to confirm theswapped expression ismeaningful.
 Let children confirm
subtraction does not havethe swapped expression toextract curiosity.

3

Teacher: Can we swap then again? 2 - 5 = ?
Children: The expression “2 - 5” does not exist.
Teacher: Aren’t we able to swap the tapes? What does “2 - 5” mean?
Children: The length difference between the 2 m tape and the 5 m tape.
Teacher: Do we know the value difference between them?
Children: The value difference is 3 m.
Teacher: The answers of “5 - 2” and “2 - 5” are same, aren’t they?
Children: These different expressions must have different answers.
Teacher: Can we describe “2 - 5 = ?” using the number line?
Teacher: Where should we place 0 this time?
Children: If we place 0 at the end of the 5 m tape, the 2 m tape is 3 m shorter
than 0.

30Children: The answer of “2 - 5” is 3 less than 0.

2

 The similarity between“2 - 5” and “5 - 2” is thevalue difference of “3”.
Teacher should intentionallymake a mistake like “Theanswer of 2 - 5 = 3.” Thisintentional wrong answer
might ignite debate amongchildren.

 Ask children the positionof the 0 and let them find 3
exists left side of the 0.