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.
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 = .....
3x + 5 = x + 4
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.”
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
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 size of the green, blue and orange shapes
Notice that the expanded form starts with 1 eg 63 = 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.
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 24 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 34 look like?
Use Dienes Blocks on MathsBot below to create your own base blocks.
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 x3
From Number to Algebra using Dienes Blocks, Base Blocks and Algebra Tiles
Compare the Dienes blocks with column addition (in a place value table).
123 + 238
Compare the example above with algebraic addition below.
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
4 x 23 = 92
Multiplication in base x