Using Money to Subtract

Introduction - Money, the Great Motivator

 

A great reason to be careful and knowledgeable with mathematical calculations is that it might cost you money one day. If you're not sure about how to add and subtract 2- or 3-digit numbers, it might cost you 2- or 3-digits worth of cash.

That was the reason I gave the kids when we launched into a bit of 3-digit subtraction work.

Money is real. People use it every day. And having 1 dollar coins, 10 dollar notes and 100 dollar notes is a great advantage when working in a base-10 number system. Probably why we don't have $6 notes or 17c coins.

Though when I was young I did see a 3c coin made by putting a 1c and 2c together on a railway track.

And blow me down - there was a 3c coin made in the USA from 1851 to 1889!

 

Setting Them Up

 

Kids were divided into small groups. Each group was given an amount of "cash":

       

10 x $100 notes

       

18 x $10 notes

      

 18 x $1 coins

 

The sample question that we were going to investigate was: 

       

 $255 - $178 = ________

 

So, I have $255. 

My greedy friend wants $178.

Let's start in the units. I have to give my greedy friend $8 - but wait! I only have $5 in coins! I need to get to the bank and change a $10 for some coins.

Phew! Now I have $15 in coins and can give my greedy friend the $8 in coins that he wants.

Now, on to the 10's column. My greedy friend wants $70 but I only have four $10 notes left! Oops - back to the bank! I'll trade a $100 for ten $10 notes.

Done. Now I can give my greedy friend $70 in notes. And he wants the $100 from my stack.

I give him the $100 note I have left. Now he has the $178 he wanted. I have $77 left.

Conclusion - $255 - $178 = $77 

And using our money we can reinforce the process of subtracting with regrouping without needing to get involved in writing anything down. Yet.

Something to Ponder

With all the trips to the bank and back, I ended up with $100 x 1, $10 x 15 and $1 x 15

but

the amount of money I had never changed (until I gave it to my greedy friend)

$255 is the same as ($100 x 1) + ($10 x 15) + ($1 x 15)

 

Going Further

A question to leave the kids with....

 

               ....why did we get 18 each of the $1 coins and $10 notes?

 

 

Playing with Squares

Introduction

We were continuing our investigation into 2D shapes. Our next focus was on squares and what we could do with them.

Challenge 1

How many different ways can you arrange 3 squares?

• Rule - they had to join by a complete side, not by corners or part of a side

Here's a solution as discovered by one of the students:

Significant discussion ensued about flips and turns. 

Were these shapes all different or just translations of the same shape?

Challenge 2

How many different ways can you arrange 4 squares?

• Same rules apply

This is what we got. Note that we had by now eliminated duplications of the same shape in different orientations.

Challenge 3

Now try it with 5 squares.

A bit more difficult but we think we found them all

Challenge 4

Find the pattern.

If 1 square can have 1 solution

   2 squares has 1 solution

   3 squares has 2 solutions

4 squares has 5 solutions

5 squares has 12 solutions....

What comes next?

Playing with Quadrilaterals

After the Triangles We Followed Up With Quadrilaterals

Question was - what shapes can you make using a pair of quadrilaterals? I was most interested in following up the comment from Ross Mannell (see "Playing With Triangles" post) who suggested that it is interesting to see if kids can make odd sided shapes from quadrilaterals.

So, we got the cardboard rectangles and the scissors out and had a play.

First reaction was - "No way. You can't make a 5 or 7 sided shape from a pair of quadrilaterals!"

But after a bit of thinking, here's what we came up with:

A Pentagon:

"Excuse me Mr Ferrington,

I think I've made a pentagon!"

A Hexagon:

A hexagon with nicely labelled sides

A Heptagon:

We explored the suggestion from Ross Mannell

to use parallelograms for this one

An Octagon:

One of the first we discovered

A 32-gon:

Just a bit of whimsy but support for the theory

that the total maximum possible sides

equals number of shapes x number of their sides

In the conversations with the kids I was careful to always say "quadrilaterals" and avoid leading their thinking in any particular direction by suggesting they stick with squares or rectangle.

Lots of fun in Year 4 this week. 

Playing With Triangles

Where did we start?

As we were exploring 2D shapes, we got some coloured card cut into rectangles. 

First job was to rule a diagonal line across the rectangle as a cutting guide. After cutting down the line we produced 2 new shapes - a pair of triangles - Viola!

Some discussion ensued - What type of triangles were they? Were they both the same? What do you mean by same? What types of triangles are there? etc

Next challenge was to see what shapes we could make with the triangles. Here's a few pictures of our results.

We found that we could use all 4 triangles to make shapes from a quadrilateral up to a dodecagon. 

- no overlapping allowed

- had to join at an edge not at a corner

4 Sides

5 Sides

6 Sides

7 Sides

8 Sides

9 Sides

10 Sides

11 Sides

12 Sides

Then we went backwards...

So we worked out that we could make a dodecagon if we had 4 triangles.

What if we only had 3?

Hmm...so we took away one triangle and found - hey we can only make up to a nonagon!

And when we had two triangles, we could only get up to a hexagon!

Amazing moment of understanding!

"Hey! That's a pattern like the 3 times tables!" exclaimed one perceptive student.

"1 triangle makes a 3 sided shape.

2 triangles can make a 6 sided shape.

3 triangles can make a 9 sided shape.

4 triangles can make a 12 sided shape."

Get the pattern?

Parting Words

"So," I said to them at the end of the maths lesson. "What is we had started with quadrilaterals instead of triangles? What would be the pattern then?"

Or if we started with pentagons?

Or if we...?