## Table of contents

Learning multiplicative thinking is one of the challenges faced by students and teachers throughout 3rd and 4th grade, and it is a challenge that continues into later grades. In this article, we present the multiplicative thinking teaching sequence that we provide at Innovamat.

Just as we talked about additive thinking in a previous blog post, and we reviewed the capsules related to addition and subtraction, today we dive into multiplication and division.

With the help of Cecilia Calvo and the rest of the teaching team, we analyze each of the decisions behind the teaching sequence. We will also look at the content from training capsules 10, 11, 14 and 15:

→ C10 – The Multiplication Tables

→ C11 – The Area Model of Multiplication

→ C14 –Division: Distribute or Make Groups

→ C15 – The Division Algorithm

## Construction of the Tables: From the Repeated Addition Model to the Area Model

In this capsule, we extensively develop the procedure students follow to learn the times tables.

The process of learning multiplications of numbers less than 10, which we traditionally call the times tables, is slow and will take up most of 3rd and 4th grade. Students have laid the foundations for multiplicative thinking in 1st and 2nd grade when they do doubles and halves and when they count grouped objects however, it’s not until 3rd and 4th grade that we build the tables and then practice productively to automatize them.

At first, we avoid following a strategy of memorization by repetition to learn the tables. The learning process is carried out in 3 phases:

**We build**the tables- We
**analyze**them for patterns and relationships - We
**start productive practice**to automatize them.

Beyond these phases, several considerations should be taken into account.

First, we should provide the students with a situation that they recognize (counting hens’ legs, car wheels, fingers, etc.), without explicitly linking it to any multiplication tables and we should always state the number that is repeated (for example, 3 hens with 2 legs, 5 hens with 2 legs, 3 cars with 4 wheels, etc.).

We do not present the tables in the traditional ascending order (1 × 3, 2 × 3, 3 × 3, 4 × 3 …) but according to the counting skills that children should have acquired. We do this to place emphasis on building them and not learning by memorizing. In this way, we start with the 2 times table, followed by the 5 times table, and we link them to counting aloud in 2s or 5s (skills that have already been worked on a lot in first and second grade). We continue with the 4 times table, which is double the 2 times table; the 8 times table, which is double the 4 times table; the 3 times table, which is the results of the 2 times table plus one; the 6, which is double the 3 times table; the 9 times table which is the 10 times table minus one, and finally we come to the 7 times table, which has already been built with the “× 7” results of the other tables, thanks to its commutative property, and that we can further explain by using the 5 times table plus two.

As we have already said, our first contact with each of the tables is neither in an orderly nor ascending way. This is to discover diverse patterns and, later, establish relationships with other tables.

## The Rule of 0 and the Area Model

**Repeated** addition of groups with the same amount of elements leads us to a model of groupings of everyday objects that serves us for counting in the early stages of multiplicative thinking, but its usefulness is short-lived. If we do not complement it with the area model (that is, understanding multiplication as a way to count elements in a given area more quickly and linking it with the calculation of areas) it becomes difficult to demonstrate the commutative property of multiplication to students.

In terms of efficiency and when the factors are numbers greater than ten, the **rule of zero** comes into play (we want the students to understand that multiplying by 10 is like adding a 0 to the number we are multiplying, this is worked on specifically).

Once the area model has been worked on, we lead the students to the next step in the abstraction process: we present the **multiplicative diagram** as a representation of the area model. In capsule C11, we explain the process in detail.

Although the foundations of division in 1st and 2nd grade have already been established by halving and distributions, now, once the foundations of multiplication have been laid, we begin to work on division in more detail.

## Types of Situations Involving Division

We continue to insist on the idea that learning basic operations has nothing to do with learning how to use algorithms. That is why the first thing we must point out is that division **allows us to solve two different types of contexts**. Although the operation carried out is the same, the context in which it is applied is not, and this is relevant when it comes to understanding the process.

**Partitive Division**: Distribution situations in which we know the number of elements that we must distribute and the number of groups, and the quotient is the number of elements in each group. For example: I have 27 pencils to distribute among 4 people.**Quotative division**: Grouping situations in which we know the total number of elements that we must group and the number of elements that each group must have, and we want to know how many groups we can make. For example: I have 27 pencils and want to make packs of 4 pencils.

## The Importance of the Remainder

Another idea which we delve into in the capsules is the **importance of the remainder**. In the everyday situations that we put forward, it appears naturally. So much so, that the remainder can be the answer to a situation in which we are asked how many of the elements have not been put into groups or distributed, for example. The concept of the remainder will also allow us to establish connections with divisibility during 5th and 6th grades.

## Construction of the Division Algorithm

We recognize that the algorithm is a powerful tool, but we know that if it is presented too early it produces cognitive passivity and interferes with the development of skills necessary for mental math. Therefore, before the presentation of the algorithm, we base calculation on the relationship between division and multiplication, **working with Multiplication Triangles** and developing other calculation strategies such as deductions based on **known facts and derived facts**.

Lastly, we explore the decomposition of the dividend into factors, which will not always be a decimal decomposition, but will be a division depending on the **divisor**. All this work is essential in constructing the algorithm solidly.

So, we base the construction of the algorithm on the **distribution model**. We insist on the idea of distributing quantities, and not digits, with agreed upon and flexible distributions.

We like to see how working this way, there is **no discontinuity between division with a one digit-divisor and a two digit-divisor**, and there is no discontinuity in the jump towards decimal division either.