Middle school maths, year 2 · unit 2
Algebraic Expressions
What is 2³ × 2²? That is 2 multiplied 3 times and then 2 more times, 5 times in all: 2⁵. Working with expressions is a matter of counting like this. We start by joining blocks into one row.
Multiply, and they join into one row
A board to arrangea³ means a multiplied 3 times. What happens when you multiply powers together? Drag the groups of blocks down onto the tray and join them up.
Divide, and they cancel in pairs
A board to arrangeA division can be written as a fraction. Drag the blocks above and below the line onto each other to cancel them in pairs.
Width times height
A board to adjust3a × 2b is the area of a rectangle 3a wide and 2b high. Drag the gold handle to make the rectangle.
Share it out to every term inside
A board to arrangeA minus or a multiplier in front of brackets has to go to every term inside. Drag the gold card onto each term card.
Try it yourself
Use what you found on the boards. The numbers change every time and the easy questions come first. Get three right and the lamp lights.
What to take away from this unit
- aᵐ × aⁿ = aᵐ⁺ⁿ: the multiplications join into one row, so add the exponents. (aᵐ)ⁿ = aᵐⁿ: the same group several times over, so multiply the exponents.
- For aᵐ ÷ aⁿ, cancel in pairs above and below. More above: a power of what is left. The same: 1. More below: it stays in the denominator.
- To multiply single terms, multiply numbers with numbers and letters with letters: 3a × 2b = 6ab, 3a × 2a = 6a².
- A minus in front of brackets changes the sign of every term inside: − (a − 4b) = − a + 4b. Then collect like terms.
- A single term times an expression multiplies each term in turn (the distributive law): 2a(3a + b) = 6a² + 2ab. Removing brackets like this is called expanding.