Powers are not easy.
They’re the first operation where there are implied operations. A power is a position of two numbers rather than a symbol. This is a ridiculous idea if you’re a kid. Why isn’t there a symbol for it? Why does the shape and size of numbers affect what they mean?
To start with, the slide below walks through the language used within powers. The grammar and syntax of mathematics communication effects our spoken language as much as our written work, and making that explicitly clear gives pupils a fighting chance at the underlying concept.
This is animated in the slides as shown below in the gif:
The next discussion prompt is to ask pupils to clap their hands when they think the representation goes from showing thirty two to three the power of two (or three squared). This is fascinating as pupils tend to not agree!
Written maths are still representations even if they’re abstract ones.
The task below asks pupils to categorise each expression in the rounded boxes by the three categories on the left. I toyed with the idea of making this a Venn diagram, but actually I felt that this would needlessly complicate things.
I was also careful to make sure that the third box had the word currently in it because even though we can’t make meaning of these symbolic representations yet, that doesn’t mean they are meaningless.
(This type of pedantry is why I don’t get invited to parties.)
The tasks below really hammer home this idea of translating between language and digital representation, particularly drawing attention to the difference between bases and exponents in similar looking cases.
When looking at physical representations of squaring and cubing, the digital manipulatives on toytheatre are great. (https://toytheater.com/cube/)
This was layered with the slide and animation below as a prompt for discussion
“Does this represent three squared or three cubed?” was asked at every new figure.
Pupils really struggle to recognise that the whole top row are still showing three squared. BUT THEY’RE MADE OF CUBES. For maths teachers we know that the width of one in that third dimension is the multiplicative identity (giving us 32 x 1 rather than 33), but this leap is significant for children and should be made explicit.
The task below involves recognising and using simple square and cube numbers. This task is designed to interleave this new information alongside powers of ten multiplication and division and long multiplication without a calculator.
While pupils have seen divisions written like the question at the end, this is a temperature check ready for order of operations stuff that comes after this in our sequencing.
The next two tasks are some of my favourites. I made these to encourage reasoning and thinking without calculation, and to begin to expose them to the idea that we can work around knowing answers. (hello algebra. hello sneaky quadratics in disguise.) These always set the scene for prime factorisation later on too.
At the start of this post I said that powers we difficult.
Thinking about square roots is also difficult.
It’s the first operation kids meet that isn’t a binary operation. This is further complicated by introducing powers as repeated multiplication – shouldn’t it’s inverse be repeated division?
I don’t think you can overdo reasoning and work with roots.
The rest of the tasks in this post aim to explore this difficulty in different ways while still giving pupils time to practice and develop their fluency in a way that doesn’t conceptually overload them but keeps consistency for surds later on.
To give pupils time to practice and really nail the recognition of their first 15 square roots, the task below was given to interleave place value understanding. Students were guided to square their answers to test out their thoughts. These have been carefully selected to ensure a rational answer.
I like these questions because they really made me think carefully about place value and my assumptions too.
The Desmos sheet here: https://www.desmos.com/calculator/vkqondn6w8 is a visual tool for showing the square roots between integers as a lead in to the next exercise.
The next task and demonstration is to estimate square roots. This recaps inequality notation and worked through examples on in the slides at the top of the post. Here, some brief discussion around why we know we don’t need less than or equal too signs helped us refresh our memory about previous learning.
The task below provides a lovely chance for some sophisticated reasoning. The first question sets up the concept of the vinculum extending beyond a single number as a grouping structure that we explore more thoroughly in the order of operations section (which will be linked here when I get round to doing it).
For example, in the top right question, if pupils can recognise that root thirty is less than six, and root five is going to be more than two, then can reason where the inequality will go.
And here is where it gets MEATY.
The animation and slide shown below outline a sequence of assumptions and checking about the distances between two consecutive square roots.
The image prompt below comes from the Desmos sheet found here: https://www.desmos.com/calculator/xwqldqncvn
While a Desmos sheet is pretty blank, the sequencing below was how I used it in lessons with year 7.
“The number line shows the square root of one and the square root of 9. What number is the midpoint of one and nine?” (Further discussion and examples of resources that focus on midpoints can be found here.)
“Where do you expect the square root of five to appear on this line?”
“Why will it not be on the 2?”
“Let’s decide before I click it to show where it is.”
“Were you right, were you not, why do you reckon this is, etc. etc. etc.”
“Where does root 2 go then?”
“What would I need to type in under my square root to get a cross on 2.5?” (Cue kids squaring 2.5 only to be UTTERLY affronted when I show them I’ve typed 2.5 squared into Desmos.)
And so on and so on.
Where this really gets into its own is in discussing the distances between the roots.
“What do you think is happening to the distance each time?”
“Will this go on forever?”
“What will happen when I click the table of values below to plot the square roots of the first thirty integers?”
This led to a really rich discussion about infinity, and how the gaps in the Desmos sheet stay the same but the gaps between the square numbers get bigger.
In terms of cube roots, I focused on the difference in symbols first.
To do this, we looked at what we mean by implied meaning in maths.
In each case above there is redundant information that we can safely assume if it’s not there.
This is the same with root symbols.
The small two in the root symbol is implied.
This is something we’ll revisit when we get to negative numbers as well, where we’ll have the language to discuss that the square root symbol implies the positive root for even powers too unless there is the plus minus symbol present. (We love a robust explanation that carries forward.)
It was also worth explicitly ensuring pupils were aware of the distinction between the three expressions shown below.
Coming off the back of division, pupils were mistaking root symbols for divisions, particularly in their own writing of questions with the cube roots. This is where I’ve asked kids to write their questions in their book using double spacing in 0.5cm squares because otherwise it’s a nonsense. This explicit look at the accuracy around their written mathematics correlates with the approach taken at the top of the post.
Finally (!) the two tasks below provided some practice for recognising and finding roots without the symbols but linked to geometric vocabulary and unit measures.
Again, here there are links to be made with place value in the bottom two of the second page, and the representation of the cube shown face on in the second page can cause confusion and is a good litmus test to see if pupils are actually reading the units or just recognising their square and cube numbers.
All resources are linked at the top of the post with solutions.