Lesson video

Hello, my name is Mr. Tazzyman, and I'm gonna be teaching you today's lesson from the unit that's all about equivalence, compensation and addition.

I hope that with some of the learning that you experience today, you'll have a better understanding of some of the underlying structures for addition problems that you face in maths.

Here's the outcome for today's lesson then.

By the end, we want you to be able to say, I can use equivalence and compensation strategies to solve problems in a range of contexts.

These are the keywords that you are gonna see and hear during the lesson.

I'm gonna say them and I want you to repeat them back to me.

I'll say my turn, say the word, and then I'll say your turn and you can repeat it.

Ready? My turn, budget.

Your turn.

My turn, unknown.

Your turn.

My turn, adjustment.

Your turn.

My turn, redistribution.

Your turn.

Here's the meaning of the words.

A budget is the total amount of money that you have to spend on something.

It is often split into parts, and money can be redistributed between these parts.

An unknown is a quantity that has a set value, but it is represented by a symbol or letter, and in this lesson, it's often part of a budget and is represented in a bar model.

When you adjust, you make a change to a number.

In this lesson.

When one part of a budget is adjusted and the other parts remain the same, then the overall budget will need adjustment.

Redistribution is where some of one part is moved to a another part is sometimes known as the same sum rule.

Here's the outline then.

Use equivalence and compensation strategies to solve problems in a range of context.

For the first part of the lesson, we're gonna be looking at budgeting, and in the second part of the lesson, we're gonna be looking at budget increases.

Let's start with budgeting.

With two friends with us, Alex and Aisha.

They're gonna help us by discussing some of the maths, presenting some of the context, and giving us some hints and tips along the way.

Hi, Alex.

Hi, Aisha.

Are you ready? Let's begin.

Oak Academy is getting a new playground.

The pupils are pairing up to create designs for the playground with a budget of 10,000 pounds.

They have to choose how much of the budget to spend on each zone.

There's an outdoor classroom, a garden, a pond, and a play area.

Alex and Aisha are working out there budgets.

Can you see they're drawn a bar model there? They've got the total at the top of 10,000.

Then they've got the outdoor classroom, 3,999, the garden, 1,350, and the pond and the play area has been written as an unknown.

What do you notice? "I don't know how much we have left to spend on the pond, and the play area says", Alex.

"It's an unknown at the moment." Alex and Aisha set about calculating the unknown.

I'll write it as an equation to help us.

3,999 pounds plus 1,350 pounds plus an unknown is equal to 10,000 pounds.

We can start by finding out the total of the outdoor classroom and garden.

Let's use a written method.

Is there a more efficient way? What do you think? Alex uses redistribution.

I'm gonna redistribute from one part to another, so he redistributes one pound from 1,350 pounds to 3,999 pounds.

That gives him a new transformed calculation of 4,000 pounds plus 1,349 pounds.

That seems a little bit easier to calculate.

Now, I've transformed the addition into something simpler.

The answer is 5,349 pounds.

Aisha writes, the new figure on the bar model, 4,000 pounds plus 1,349 is equal to 5,349 pounds.

"Wait! I'm not sure that's right." What has Alex seen? Think carefully about what they were calculating.

That combined total was for the cost of the classroom and garden.

We still need to calculate the pond and play area.

"Oh, yes.

Columns?" "I think we can do it mentally still." Alex uses equivalence to help.

"I'll look at the parts and start with a known fact that's close to the equation we are solving." 5,350 pounds plus 4,650 pounds equals 10,000 pounds.

"I used my knowledge of place value and complements to 1,000.

Then I'll write it out the equation we want to solve underneath to compare." 5,349 pounds plus an unknown is equal to 10,000 pounds.

The first part has decreased by one pound, but the whole is the same, so our unknown is one pound more than the part in the known fact.

The answer is 4,651 pounds.

Aisha writes the new figure on the bar model.

There it is.

"Now we need to decide how much to spend on the pond and how much to spend on the play area.

I really like ponds, so I think we should spend 4,000 pounds on that." "How much would that leave to spend on the play area?" I'll write an equation again.

3,999 pounds plus 1,350 pounds plus 4,000 pounds plus an unknown is equal to 10,000 pounds.

It leaves 651 pounds because we have partitioned.

Eep.

That's not a lot on the play area.

Aisha and Alex realise that they have missed something.

Play area, minimum spend 2,000 pounds.

"What does this mean?" "It means that we have to spend at least 2,000 pounds on the play area to get the right equipment, so we need to adjust our budget then." "How much do we have to adjust by" says Aisha.

"651 will need to be adjusted to 2,000." "Surely a written method here?" "No! We can use complements to 1,000." 651 plus 350 is equal to 1000.

"Now I'll redistribute." 651 plus 349 is equal to 1,000.

"I get it! Now we can adjust the hold to 2,000." 651 is staying the same, so 349 will need to be adjusted because the expressions are equal.

We had to increase the spending by 1,349 pounds.

Aisha amends the bar model and rewrites the equation.

There's the bar model with the play area now costing 1,349 pounds, and then she writes out the equation below.

"Respectfully, I don't think this is correct", says, Alex.

What do you think? Is there something Aisha's forgotten about? She's written in the new value for the play area, the new cost, but is there something else that needed to change? Alex says, "You've written in the adjustment, not what we are spending on the play area.

Also, you've increased one part but not decreased any other parts, so we are over budget!" "Let's show that with a greater than symbol.

In the bar model, I'll write in the adjustment." "How can we get back under budget then?" What do you think? "We have overspent by 1,349 pounds, so we need to reduce the other parts by 1,349 pounds altogether." Aisha and Alex both come up with a new budget by reducing a part by 1,349 pounds.

There's Aisha's and there's Alex's.

What's the same? What's different? "I reduced the outdoor classroom part", says Aisha, "I reduced the pond part", says Alex.

"The whole is 10,000 pounds." "We both have all four parts." Okay, here's your first practise task.

Look at the budget equations below.

Can you work out the unknowns without using a written method? The first one is complete already.

That's probably gonna help you out.

Here's number two then.

Look at the budget in the bar model below.

Without using a written method, calculate the unknown.

For number three.

Below are some minimum spends for each zone.

10,000 pounds is still the budget.

Is this possible? Explain your reasoning.

For number four, look at the budget in the bar model below, there are some missing digits.

How many ways can you complete the missing digits? Okay, pause the video here and have a go at those.

Enjoy.

Welcome back.

Let's start with number one then.

The missing numbers.

First it was 3,651 pounds.

Then it was 1,000 pounds, 700 pounds, 3,581 pounds.

Here's number two then.

I noticed that 325 and 675 are complements to 1,000.

That meant that 2,325 pounds and 2,675 pounds is equal to 5,000 pounds, so the pond and play area was 5,000.

Then I used a known fact and redistributed.

2,500 plus 2,500 equals 5,000.

If we add 51 onto that first part, we get 2,551 plus 2,449 is equal to 5,000, so the garden is 2,449.

Let's have a look at this.

Alex is gonna explain what he thinks.

"It's impossible!" He says.

"The four parts combined are more than 10,000 pounds." And number four, looking at the budget below and thinking about the digits, Alex is gonna explain as well.

"I knew that the ones column needed to be a zero to match the budget.

That meant that the total of the ones digit had to be 10 or zero.

There was already a two in the ones column of the pond part, so the missing ones digit had to be eight.

The classroom and garden part added together were equal to 5,050 pounds, which left 4,950 pounds.

If both the missing digits were zero, then the parts would combine to make 4,810 pounds.

That would be 140 pounds less than the budget, so the tens digit combined needed to be equal to 14 because 14 tens are 140 pounds.

Nine and five, eight and six, seven and seven, six and eight, five and nine." Okay, let's move on to the second part.

The lesson now tallies and bar charts.

Ready? Let's go for it.

The playground budget is finalised.

The work begins.

You can see there's 10,000 pounds in total.

The outdoor classroom budget is 3,000 pounds.

The garden is 2,000 pounds.

The pond 1,500 and the play area 3,500.

"It's very noisy outside", says Aisha.

"Pardon? Says Alex." Oak Academy gardening club runs a fundraiser to help reduce the cost of the garden zone.

They sell fresh vegetables they have grown in a nearby allotment.

Onions, spring onions, potatoes, and radishes.

Delicious.

The tally chart below shows how many of each vegetable they sold from Monday to Thursday.

Alex calculates how many vegetables were sold altogether, onions, spring onions, potatoes, and radishes, and they've got the tally chart next to it.

Let's start with each tally.

27 in the first, 38 in the second, 12 for the potatoes and 23 for the radishes, and remember Alex was able to do that really quickly by counting in fives.

Each time that there's a horizontal stroke across a tally of four, you know that that counts as five.

"I've spotted something," says Alex.

What has Alex noticed about the numbers? "There are pairs of complements to 50." 27 and 23 and 38 and 12.

That means there were 100 vegetables all together.

On Friday, they sold three more onions, five more spring onions, and one more potato.

Mr and Mrs. Dour, the school neighbours brought back four radishes for a refund saying they were too spicy.

Alex calculates the new total number of vegetables sold.

"I'll start by writing down the changes." Three more onions, five more spring onions, one more potato, four fewer radishes.

"Then I'll recalculate each part." "You don't need to," says, Aisha.

What do you think? Do you think Alex should recalculate the total for each of the vegetables? You're right.

I'll just work out the adjustment", says Alex.

Three plus five plus one takeaway four is five, so they know in total they sold five more vegetables.

The total was 100, and now I'll adjust that.

100 plus 5 is equal to 105.

After the new playground is completed, savings and extra costs emerge.

The bar chart below shows those savings and extra costs.

"Which ones are extra costs and which are savings?" Good question, Aisha.

It can be a bit confusing when you first see a bar chart like this.

"Well, the garden was our fundraiser and we made 200 pounds." "So that can be subtracted from the total cost?" "Yes, definitely.

So can the pond.

But the classroom and play area cost more." "I'll label the chart to help us remember." Extra costs, savings.

That's much clearer.

Well done, Aisha.

Alex and Aisha have been asked to work out if the playground costs more or less than the budget of 10,000 pounds.

"Let's work out the adjustments to each zone cost first." "The outdoor classroom had an extra cost of 400 pounds.

The garden had a saving of 200 pounds.

The pond had a saving of 450 pounds, and the play area had an extra cost of 200 pounds." "Let's calculate the cost of each part." "Do we need to?" What do you think? Do you think they need to calculate each area again? "If you adjust the parts, then the whole adjusts by the same amount." "Okay, here's the expression then." 10,000 pounds plus 400 pounds, subtract 200 pounds, subtract 450 pounds plus 200 pounds.

"Two of the parts cancel each other out." Now we're left with 10,000 pounds plus 400 pounds.

Subtract 450 pounds.

Let's partition 450 into 400 and 50.

More parts cancel each other.

9,950, so it was under budget.

Okay, your turn to do some tasks.

For number one below is another tally chart for vegetables sold from Monday to Thursday at a farmer's market.

The next day, they sell 6 more carrots, 10 more tomatoes, and 2 more radishes, but they refund 11 lettuces because they have bugs in them.

How many vegetables have they sold all together over five days? Here's number two.

Here's the budget and costs and savings bar graph for a larger school's new playground.

You can see the bar model there.

That's got the budget in it with the four different zones that you were learning about earlier, and there's a new bar chart there as well.

You've got to use mental methods and jottings to solve the following.

A, what was their budget? B, how much did it actually cost? And C, did they overspend or underspend by how much? Use the information there and take care as you're going through working out each of those.

Okay, pause the video here and I'll be back in a little while with some feedback.

Good luck.

Enjoy.

Welcome back.

Let's start with number one then.

To begin with, we'll look at what the totals were for each of those vegetables.

23 carrots, 29 tomatoes, 11 lettuces, and 22 radishes.

If you look closer, you can see 29 and 11 combined add up to 40, and 23 and 22 add up to 45, so that meant that there were 85 altogether.

There's the adjustments listed down, 6 more carrots, 10 more tomatoes, 11 fewer lettuces, and 2 more radishes.

All together, that means we have 85 plus 7, which meant 92.

Here's number two then.

A, what was their budget? Well, we were adding together all four of the costs of those zones.

You might have noticed that 360 and 640 were complements to 1000.

That meant that if you'd added together the play area and the garden, you'd end up with 8,000.

4,000 plus 1,500 was 5,500, so in total their budget was 13,500.

B, how much did it actually cost? Well, if we list down the adjustments, we can see there were 250 pounds for the outdoor classroom extra cost.

300 pounds for the garden extra cost.

A saving of 250 pounds of the pond and a 200 pound extra cost for the play area.

Altogether that meant 500 pounds extra, so it cost 13,500, which was their budget, plus an extra cost of 500 pounds, that's 14,000.

Did they overspend or underspend by how much? Well, we know the budget was 13,500, but the cost was 14,000, so they overspent by 500 pounds.

Okay, that's the end of the lesson, and here's a summary of the things that we've learned about.

Equivalence and compensation are useful tools for simplifying problems and helping us to find unknowns efficiently.

Adjustment and redistribution work from these principles.

This can be applied when considering budgets and costs.

It is also very useful when analysing increases and decreases from statistical data.

My name is Mr. Tazzyman.

I've really enjoyed today's lesson.

I hope you have as well.

Bye for now.