# Number to Algebra

Base blocks make the link between Dienes blocks and algebra tiles by exposing the mathematical structure of the place value of the base system. In this section we will look at the place value structure of the base system using counters and the geometrical structure using blocks. Both structures are important to gain a secure and deep understanding of the base system, including base 10 (Dienes blocks) and base x (algebra tiles).

## A coherent journey through the base system

**Dots and Boxes**** **

**Dots and Boxes**

The content is this section is adapted from https://www.explodingdots.org/ by James Tanton.

This set of examples lets students explore place value in different bases, leading to a deeper understanding of base 10. This can be applied to addition, subtraction, multiplication, standard form and negative indices. To go even deeper you can work in base x. I use this to introduce algebra tiles.

**Base 2**

**Base 2**

**The 1 ←2 rule:** Whenever two counters are in a box together, they stick together and move one place left. This is a plan view so you cannot see how many counters are in each pile on these diagrams. Try this activity with counters or multilink cubes to get a real feel for it.

At each stage count up all the counters in the boxes and below and you should have 9.

Two counters in the right hand box and 7 below.

These two counters stick together and move one place left.

Now this box has a pile of 2 counters and 7 below.

Add two more counters to the right hand box…

There are now 4 counters in the boxes and 5 below

The two counters stick together and move one place left.

This is a pile of 4 counters and 5 below.

Keep adding counters from the right.

You should end up with a pile of 8 counters and a single counter.

The piles of counters can eventually be replaced with a place value chart.

Using base blocks or Cuisenaire rods exposes the geometrical structure of base 2.

Moving 1 place left means x2 so the opposite must be true, moving 1 place the right means ÷2.

It is useful to compare this with moving left and right in base 10, a concept that students are familiar with.

This model contains one of each base 2 block up to 2 cubed.

Get students to show different numbers in base 2. They could create their own using multi-link cubes or use the Desmos graph on the left.

Why are there only one of each block?

What would 2^{4} look like?

This model contains two of each base 3 block up to 3 cubed.

Get students to show different numbers in base 3. They could create their own using multi-link cubes or use the Desmos graph on the left.

Why are there only one of each block?

What would 3^{4} look like?

Use Dienes Blocks on MathsBot below to create your own base blocks.

Use the dynamic base blocks below to change the base.

**Negative Indices**

**Negative Indices**

### Multiplicative Opposite (Inverse)

**Multiplicative Opposite**

2^{3} = 1 x 2 x 2 x 2

2^{-3} = 1 ÷ 2 ÷ 2 ÷ 2

2^{5} = 1 x 2 x 2 x 2 x 2 x 2

2^{-5} = 1 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2

**2**^{3}** is repeated multiplication**

**2**^{-3}** is repeated division**

**Minus exponent means opposite of repeated multiplication**

3^{5} = 1 x 3 x 3 x 3 x 3 x 3 3^{-5} = 1 ÷ 3 ÷ 3 ÷ 3 ÷ 3 ÷ 3

5^{5} = 1 x 5 x 5 x 5 x 5 x 5 5^{-5} = 1 ÷ 5 ÷ 5 ÷ 5 ÷ 5 ÷ 5

Base 10 brings us back to Dienes blocks. Most students will be familiar with these from Primary school.

Using base blocks exposes the geometrical structure common to all bases.

### Base x

Algebra tiles can be used to represent the area model for multiplication and division and much more.

**Variable Base Blocks (-10<x<10)**

**Variable Base Blocks (-10<x<10)**

Drag the green x bar below to reveal the geometric and place value structures of base -10 to 10.

Try changing the base with these Dienes Blocks in MathsBot. What structures does it reveal?

Working with Base Blocks in different bases leads to base x. Algebra tiles on MathsBot show the geometric structure of base x.

**Base x Blocks (algebra tiles) include x**^{3}

The model below connects algebra tiles and the grid method commonly used for expanding and factorising.

**OFSTED Research Review Series: Mathematics**

**OFSTED Research Review Series: Mathematics**

Informal methods, some of which may involve physical resources, can be useful for revealing underlying principles and concepts.^{[footnote 82]} However, teachers need to be cautious when considering curriculum approaches that are heavily weighted towards encouraging informal and self-generated methods. These approaches may purport to develop pupils’ understanding, but the evidence shows that when pupils use a variety of informal procedures, it can inhibit understanding later on.^{[footnote 83]}

Additional risks arise from mixing and matching a toolkit of informal and self-generated methods for working with larger numbers and more complex calculations as pupils progress through the curriculum. This increases the likelihood of pupils generating errors and structuring written records poorly, which may lead to confusion.^{[footnote 84]}

Teachers should seek to balance developing pupils’ understanding and its associated use of informal and diagrammatic methods with instruction in efficient methods that accurately and consistently reveal new patterns and connections of number. This is because the 2 aspects of understanding and computational proficiency reinforce and augment each other.^{[footnote 85]} One way to achieve this is to plan to use informal methods for only a short amount of time, as a bridge to formal written methods. This would ensure that pupils have adequate opportunities to learn, rehearse and then use formal methods. The earlier learning and therefore increased use of core mathematical methods^{[footnote 86]} also gives greater assurance to teachers that their pupils will be ready to use these methods within sequences of calculation and to solve more complex problems in their next phase of learning.

**Methods for working algebraically**

**Methods for working algebraically**

The message of quality over quantity of procedural knowledge also applies throughout key stages 3 and 4.

In algebra, pupils benefit from fewer but powerful representations and an iterative approach to sequencing the facts and procedures for working algebraically.^{[footnote 87]} Abstract representations can be just as effective as contextualised representations.^{[footnote 88]} The bar modelling method can be used as a bridge from arithmetic to early algebra. It is a useful interim method for abstracting arithmetic and algebraic expressions from word problems.^{[footnote 89]} Teachers can even teach methods of evaluation of algebraic expressions and ways to set these out as a series of steps for pupils to learn by heart.^{[footnote 90]} This contrasts with an approach of encouraging more informal, self-generated ways for pupils to solve linear equations. This may be self-limiting when pupils are faced with unconventional presentations of linear equations.

If pupils learn high-quality, useful and efficient procedural knowledge, they can then apply this to setting out and using formulae, from calculating areas and perimeters of different classes of polygon in key stage 2, to using the trigonometric formulae such as the cosine rule in key stage 4.

**From Number to Algebra using ****Dienes Blocks, Base Blocks and Algebra Tiles**

**From Number to Algebra using**

**Dienes Blocks, Base Blocks and Algebra Tiles**

Dienes blocks are an excellent visual representation of base 10 that allow students to see the connections with base x (base blocks and algebra tiles). You can use Dienes blocks to address common misconceptions with standard algorithms and make connections with algebraic manipulation. This section aims to make the connection between the representation and more formal written method by comparing the same mathematical structures. It is intended that students will move to the more formal written methods when the connections are secure.

The same representations are used throughout so that students can make connections across the curriculum.

### Addition

### Subtraction

The same methods can be used for base x (using – as opposite and zero pairs)

### Multiplication

### Multiplication in base x

**Students make the connections between manipulating number and algebra using consistent representations.**

### Division

### Division in base x

**Students make the connections between manipulating number and algebra using consistent representations.**