Psychoarithmetic class (7-9yo): Lesson Notes 16 Oct 2023

​Lesson Notes 16 Oct 2023
I.         Area or Perimeter?
In this activity students have to categorise descriptions given in their workbook as either ‘Area’ or ‘Perimeter’ based on what they have learned earlier. The material ‘Area and Perimeter Interactive Notebook’ was purchased on teacherspayteachers.com and sold by the ‘Not So Wimpy Teacher’ store.
An example of a perimeter question is how far is the distance around a block, and one for area is how many tiles you need to cover a bathroom floor? There are only three descriptions for each category, so this is a brief exercise that does not aim to exert the child too much but rather to impress the concept of area and perimeter and how they are distinct from each other.
It is evident from the question that a practical approach has been taken to enable students to visualise the object of focus, area or perimeter, that is represented by everyday objects they should be familiar with. In this case, bathroom tiles and distance.
Students already know the concept of area and perimeter, there is prior knowledge, and this rather plays a reinforcing or in Montessori, is a form of memorisation work.
Extension
While there is no need to make this a longer exercise than what is already provided, teachers can extend on the lesson by asking students to come up with their own everyday description of area and perimeter and ask their friends to guess which category it should be put into.
Analysis
Most students would not confuse over area and perimeter, although it may happen at the early stage. It is good to establish concepts before doing more advanced work on a topic after their introduction to avoid misconceptions that at best can cause a slip in judgement and at worst, seeing students mistaking a question on area for one on perimeter.
Clarity of concept shortens the time taken to solve a question and reduces likelihood of using the wrong approach such as in choosing which formula to use.
Dyscalculia Focus
Such exercise which deliberately points out the distinction between concepts that are different but belong to the same family reinforces concepts that have been introduced to students with dyscalculia and are helpful with:

1. Helping students decide where to place a question
2. Relating it to the required formula needed
3. Provide a firm foundation on how the question should be approached where further adaptation is needed. For example, if a triangle is cut off from the edge of my square, then to find the remaining area I would then have to subtract the area of the triangle from the square.

II.        Find the missing part of the division equation
Students have already learned single digit division for numbers to 10,000 with manipulatives such as the golden beads, the stamp game, and the division board. They have been introduced to written long division in single digit with numbers up to 1,000. They have not been introduced to test tube division.
Recently, they were taught division with remainder in single digit for number up to 100 on the division board and in long division. Today, they were taught on how to find missing components of a division equation and why we carry out the actions.
Students were first reminded of the nomenclature of the division equation:
Dividend  ÷ Divisor = Quotient
After reviewing the above we looked at two situations where the equation is used to find a missing variable on the LHS, i.e. the dividend or the divisor:
 
Case 1                         ____ ÷ 4 = 7
Case 2                             15 ÷ _____ = 3
Case 1: Using the example of one of my students and her three siblings, I had asked if she received $7 after the money her grandmother gave her and her siblings were divided, what did she receive at first? We note that the quotient shows only what one person receives or in general how much there will be in every divisor. We place seven beads on the division board. If the division board was a table, and everyone starts putting their $7 neatly, column by column on the ‘table’ we can eventually see what was given in total. We note that this involves multiplying the rows by the column or the quotient by the divisor or vice-versa. Thus to find a missing quotient we multiply the row by the divisor.
Case 2: In the second case we do not know how many groups the quotient is divided into but we know that each group will have three members. Somehow this was more intuitive and one of the students was able to give the answer of 5. We can see that Case 1 goes from part to whole while Case 2 from whole to parts. Simply by grouping 15 items into threes we are able to find the answer and it can also be obtained by dividing our 15 beads on the division boards in threes till they run out to place five skittles on the board. Thus to find the missing divisor we divide the dividend by the quotient.
Conclusion
While both are on the opposite side of the equation, the approach taken is different for the  dividend or the divisor in relation to the quotient. While we can say the reverse of dividing the dividend by a divisor is multiplying the quotient by the divisor, Case 2 cannot be said to be the reverse but rather  a division of the divisor by the desired content in a group where the number of groups is unknown. It does not involve part-whole relationship.
 
 
Dyscalculia Focus
Dyscalculics or novice learners should not rely on simplified definitions as shown in the different approaches needed for case1 and case 2 above. If the same approach was taken as for Case 1, students may think that they should multiply 3 by 15 to get the answer of 45. This is because division is not commutative like in addition and multiplication.
This condition should be looked out for when students move to division. While their addition and multiplication should have stabilised by the time they reach division, this stable belief may influence how they treat division.