**Number to Algebra**

Algebra at Secondary school is key to further study in Mathematics and other STEM subjects but it also the area that many students start to fall behind. The abstract nature of algebraic notation and the understanding of equivalence are some reasons for this. In this section I model introducing algebraic notation and algebraic thinking as a natural extension of the number system that students are familiar and comfortable with.

### Equivalence

If students struggle to answer the questions circles (left), they could have an unsophisticated understanding of =.

Research shows that this early misconception of the meaning of = can and often will lead to difficulties with algebraic thinking.

**operational understanding ** indication of where to write the answer or carry out an arithmetic operation

3x2 + 5 = .....

**relational understanding** indicates that two sides of an equation have the same value.

3x + 5 = x + 4

**substitutive understanding** is understanding when two expressions are equal they can be used interchangeably.

y = 3x + 5 x + y =18 x + 3x + 5 = 18

=

“Without doubt one of the most important concepts that supports children’s development of algebraic thinking is the concept of equivalence.”

(Carpenter & Levi,2000; MacGregor & Stacey, 1997)

I use this example (right) in year 7 to check students understanding of equivalence.

I start with = on the board and ask students to explain the meaning of the symbol.

Students who initially say = means the answer typically struggle to answer the questions on the right.

NCETM Secondary Mastery PD material - KS2 examples

**Example and non example**

Defining the equals sign alongside these other commonly used symbols helps students to get a clearer picture of what is equal and not equal.

## From the Structure of Number to Algebra

Dietmar Küchemann (1978) identified the following six categories of letter usage by students (in hierarchical order):

**1. Letter evaluated:** the letter is assigned a numerical value from the outset, e.g. a = 1.

**2. Letter not used:** the letter is ignored, or acknowledged, but without given meaning, e.g. 3a taken to be 3.

**3. Letter as object:** shorthand for an object or treated as an object in its own right, e.g. a = apple.

**4. Letter as specific unknown:** specific but unknown number and can be operated on directly.

**5. Letter as generalised number:** seen as being able to take several values rather than just one.

** 6. Letter as variable:** representing a range of unspecified values, and a systematic relationship is seen to exist between two sets of values.

The first three categories on the left are all too familiar misconceptions that create a barrier to algebraic thinking and understanding. The rest of this section looks at ways of introducing algebra that introduce letters as generalised number and later as variables and specified unknowns.

### Place Value Structure

Place value has obvious connections to decimals, rounding, standard form etc when working on number. The place value structure of algebra is often overlooked and provides a depth of understanding of algebraic notation.

Algebra tiles (above) are the 2D version of Base x blocks (left)

**What stays the same? **

The shapes:

•yellow is always 1

•green is always a line

•blue is always a square

•orange is always a cube

The indices/powers

**What changes?**

The size of the green, blue and orange shapes

Notice that the expanded form starts with 1 eg 6^{3} = 1 x 6 x 6 x 6. This is important when you extend place value to the right of 1.

The place value structure of different bases can be generalised to base x.

The letter x is used to generalise the base system. It shows the **geometric structure** and **place value structure** of base x.

**Starting at 1 represents negative indices as repeated division.**

**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**

**Base Blocks Activities**

**Base Blocks Activities**

Introducing base blocks using multi link cubes.

More activities coming soon...

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 is 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 two of each block?

What would 3^{4} look like?

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

**Base x**

**Base x**

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

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}

**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

Compare the Dienes blocks with column addition (in a place value table).

123 + 238

Compare the example above with algebraic addition below.

### Subtraction

Compare the Dienes blocks with column subtraction (in a place value table).

238 - 154

Compare the method above to 238 + -154 using zero pairs below.

238 + -154

exchange 1 hundred for 10 tens

simplify the hundreds, tens and ones using zero pairs

The same methods can be used for base x

### Multiplication

Dienes blocks

Area model

Grid model

4 x 23 = 92

Dienes blocks

Area model

Grid model

### Multiplication in base x

Algebra tiles

Grid model

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

### Division

### Division with exchanges

### Division in base x

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