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Hi, my name is Mr. Tazzyman and I'm looking forward to learning with you today.
This lesson is from the unit all about equivalence and compensation with addition.
It'll give you some extra hint and tips on how to face addition problems and solve them mentally rather than jumping straight to a written method.
Let's get started then.
Here's the outcome.
By the end of the lesson, we want you to be able to say, I can explain how adjusting one addend or part affects the sum or whole.
Here are the key words that you might expect to hear today.
I'm gonna say them and I want you to repeat them back to me.
So I'll say my turn, say the word and then I'll say your turn and you can repeat it back clear.
Okay, let's do it.
My turn, equation.
Your turn.
My turn, adjustment.
Your turn.
My turn, whole.
Your turn.
My turn, part.
Your turn.
Here's the meaning of each of these words.
An equation is used to show that one number calculation or expression is equal to another.
An adjustment is a change to a number using either subtraction or addition.
This can be done to a part or a whole.
The whole is all the part or everything, the total amount.
A part is some of the whole.
The bar model there shows the relationship between whole and part.
Here's the outline then.
We're gonna start by looking at adjusting one part and how that changes the whole.
Then we're gonna move on to some different contexts.
In this lesson, you're gonna meet Sofia and Aisha.
They're two maths buddies who are gonna be helping us by discussing some of the maths that we are learning about, giving us some prompts, hints and tips, and sometimes revealing some of the answers too.
Alright, then you ready to begin? Let's go for it.
Here are two part-part-whole models.
What's the same, what's different? So have a look at those and compare them.
What do you think? Sofia says, "There are two parts the same in each." The ones are the same in the other part of each.
The ones are in the whole are the same as well.
In the second model, one part has 20 more, so the whole is also 20 more.
Aisha turns the models into equations.
5 + 3 = 8.
25 + 3 = 28.
What do you notice? One of the parts is adjusted.
It has been adjusted by adding 20.
Consequently, the whole has been adjusted.
It has been adjusted by adding 20 as well.
Sofia looks to investigate a pattern.
"I'll use arrow cards," she says.
5 + 3 = 8 and there are the arrow cards.
15 + 3 = 18, and she's put in 10 in the whole and in the first part.
25 + 3 = 28.
35 + 3 is equal to 38.
45 + 3 =48.
55 + 3 =58.
Can you make a generalisation here? If an amount is added to one part, then the same must be added to the whole to keep the equation balanced.
Aisha sets Sofia a challenge.
1 + 6 = seven.
"I'll start with a simple equation.
Then I'll write another equation with some missing digits in the whole.
What are the missing digits?" She's written 241 + 6 = something, something, 7.
You've adjusted one part by adding 240 and kept the other part the same.
That's the six.
So you must have added 240 to the whole.
"That means the digits need to be a 2 in the 100s place and a 4 in the 10s place." Okay, it is time to check your understanding so far.
What are the missing digits in the whole below? 4 + 1 = 5.
354 + 1 = something, something five.
Pause the video here and have a go at that.
Welcome back.
Aisha says, "One part was adjusted by adding 350 so the same adjustment needs to be made to the whole." It's 355.
Aisha and Sofia have been saving up to go and see their favourite singer.
So far they've saved 100 pounds.
Aisha has saved 56 pounds and Sofia has saved 44 pounds.
"I'll show this as a bar model," says Aisha.
You can see she's got a hold of 100 with parts of 56 and 44.
Aisha has given another 10 pounds to add.
How much have they saved altogether now? If I add 10 pound to my part, then I need to adjust the whole as well.
So we have saved 110 pounds.
Okay, here's the first practise task for number one.
Aisha and Sofia have each written two equations by adjusting apart, fill in the blanks.
You can see there that some of the adjustments are blanked out on the jottings and there are also some sentences at the bottom which have some blank for you to fill in words as well.
Number two, Jun and Jacob have also been investigating adjusting one part in addition.
Below are some of their generalisations.
Who do you agree with? Explain your reasoning and write equations as proof.
For number three, there's two worded problems to solve.
A, Aisha has 54 battle robot cards and Sofia has 66.
So between them, they have a collection of 120 cards.
Aisha has given 12 more for her birthday.
How many do they have between them now? For B, Sofia and Aisha are saving together to get a drum kit and a guitar.
Sofia has 36 pounds and Aisha has 49 pounds.
So between them they have 85 pounds altogether.
Aisha spends 17 pounds of her money on some drumsticks.
How much do they have altogether now? Pause the video here and have a go at those questions and I'll be back in a little while with some feedback.
Enjoy.
Welcome back.
Let's look at number one to begin wit.
, For the first bit, the jottings, we needed to put in 100 because that's what the part and the whole were adjusted by, and then the missing digit, therefore was a one to complete the sentence.
Then you had one part was adjusted by adding 100 and the other part remained the same, so the whole needed to be adjusted by adding 100.
Let's look at the second one now.
So this was adjusted by subtracting 200 from the first part and therefore from the whole as well, and the missing digits were a seven and five representing 75, and then we had one, seven, and nine for the new whole.
That meant that the sentence had to read as follows.
One part was adjusted by subtracting 200 and the other part remained the same, so the whole needed to be adjusted by subtracting 200.
Okay, here's number two then.
We'll start with looking at Jacobs.
He said if I adjust one part, then the whole has to be adjusted in the same way to maintain a balanced equation.
This is correct.
When one part is adjusted, then the whole needs to be adjusted to ensure the equation is balanced.
Here's an example.
3 + 4 = 7.
Three's been adjusted by adding 30, making 33 + 4, so the whole had to be adjusted as well because 33 + 4 = 37.
Let's look at Jun's now.
"In order to maintain a balanced equation, if I change one part, then I have to do exactly the same adjustment to the other part." This is incorrect.
If we adjusted both parts by the same amount without adjusting the whole, then the expressions would be imbalanced.
Here's an example.
3 + 4 = 7.
If we adjust three by adding 30, but also adjust 4 by adding 30, we end up with 33 + 34, which is not equal to seven.
Let's look at number three then.
A first.
Here it is as a bar model, the whole is 120 and the parts are 54 and 66.
12 was added onto Aisha's part, so 12 needed to be added onto the whole as well, giving 132.
They've now got 132 cards between them.
Here's B, the guitar and drum kit.
You might have started this as an equation which read 36 pounds plus 49 pounds is equal to 85 pounds.
Then we had 36 pounds plus 49 pounds, take away 17 pounds, which Aisha spent on drumsticks is equal to 85, take away 17, giving a total of 68 pounds.
Okay, let's move on to the second part.
Some different contexts.
Sofia investigates adding decimal fractions.
"I'll use place value count," as she says, 0.
5 + 0.
3 = 0.
8, and she's laid out that equation as place value counters.
1.
5 + 0.
3 = 1.
8.
Can you see how she's changed the place value counter arrangement? She's put in ones, 2.
5 + 0.
3 = 2.
8, 3.
5 + 0.
3 = 3.
8, 4.
5 + 0.
3 = 4.
8, 5.
5 + 0.
3 = 5.
8.
What do you notice? Our generalisation works for decimal fractions too.
If I adjust one part with addition, I have to adjust the whole with the same addition.
Aisha investigates subtracting from a decimal fraction part.
"I'll use place value counters." 4.
5 + 0.
3 = 4.
8.
There it is using place value counters.
3.
5 + 0.
3 = 3.
8.
She subtracts ones.
2.
5 + 0.
3 = 2.
8, 1.
5 + 0.
3 = 1.
8.
Our generalisation works with adjustment through subtraction here.
If if I adjust one part by subtracting an amount, I have to adjust the whole by subtracting the same amount, even if it's decimal fractions and a decrease of the part.
Aisha and Sofia both have pet cats.
They weigh them and find out that the cats have a combined mass of 10 kilogrammes.
"I'll show this as a bar model." 10 kilogrammes as the whole.
One cat weighing for 4.
7 kilogrammes, the other with a mass of 5.
3 kilogrammes.
A month later, they discovered that Aisha's cat has gained 0.
9 kilogrammes.
What's the combined mass of both cats now? What do you think? Let's start by working out the weight of my cat.
Sofia says, "I don't think we need to!" "You are right! We can just adjust the total." Exactly.
10 + 0.
9 is= 10.
9.
So now the total is 10.
9 kilogrammes.
Time to check your understanding.
Two dogs have a combined mass of 20 kilogrammes when weighed in January.
A month later, the larger of the dogs has lost 0.
75 kilogrammes.
What is their combined mass now? Okay, pause the video and have a go.
Welcome back.
If one dog has lost 0.
75 kilogrammes, then the total mass needs to be reduced by 0.
75 kilogrammes.
I know that one subtract 0.
75 is equal to 0.
25, so I know that 20 subtract 0.
75 is equal to 19.
25.
So that's the answer.
19.
25 kilogrammes.
Aisha investigates larger numbers.
I'll use place value counters.
Can you see that the place value counters have different values now? These are hundreds, 500 + 300 = 800.
1,500 + 300 = 1,800 and she works through.
Our generalisation works for larger numbers.
If I adjust one part, I have to adjust the whole by the same.
Sofia investigates subtracting from larger number parts.
"I'll use place value counters." 4,500 + 300 = 4,800.
There, that is as place value counters.
Can you see what happened? She keeps going.
What do you notice? "Our generalisation works with adjustments through subtraction here.
If I adjust one part by subtracting an amount, I have to adjust the whole by subtracting the same amount, even if it's larger numbers and a decrease of the part." Aisha and Sofia play an arcade game as a pair.
Aisha scores 13,470 points and Sofia scores 15,530 points.
Their total score is 29,000 points.
"I'll show this as a part-part-whole." There's Sofia's score.
There's Aisha's score.
That's their total score together.
After they both lose all their lives, they realise that Sofia has been hit by a penalty reduction of 1,750 points for pressing the buttons too hard.
"I'll show the change on the part-part-whole model." Can you see there? That's 1,750 has been subtracted.
"Remember, we don't need to calculate the part.
Only the whole." "I'll partition 1,750 into 1,000 and 750.
29,000 subtract 1,000 is 28,000.
I know that 1,000 subtract 750 is equal to 250.
So 28,000 subtract 750 is equal to 27,250." Okay, time for you to have a go at doing some workings yourself.
They have another go at the game.
They get a combined score of 30,000.
This time Aisha gets a penalty reduction of 2,650 points for holding down the buttons.
How many points did they score after the reduction? Pause the video and have a go at that.
Welcome back.
Here's what you might have done.
You can see that there's a reduction on Aisha's score, which gives a reduction on the whole.
"I'll partition the adjustment into 2,650," and you should get 27,350 points.
Okay, time for your second task.
Starting with the following equations, write your own adjustment of one part and then describe the changes in a written paragraph using our generalisations.
Two, Izzy and Alex have also been investigating adjusting one part in an addition with decimal fractions and larger numbers.
Below are some of their generalisations.
Whom do you agree with? Explain your answer and write equations as proof.
Izzy says, "If I adjust one part, then the whole has to be adjusted in the same way to maintain a balanced equation.
This is true for decimal fractions and larger numbers too!" And Alex says, "In order to maintain a balanced equation with parts contained decimal fractions, I can only make adjustments by adding or subtracting decimal fractions." And here's number three.
The table below shows the scores of different pairs playing the arcade game.
Most points wins.
There's also a list of penalty reduction and bonus addition points.
Can you order the pairs placing their scores in first, second, and third? Pause a video and have a go at those tasks.
Good luck.
Welcome back.
Let's start with marking number one then.
Be ready.
We chose 270 is our adjustment for the first equation.
So the new equation read 282.
6 + 1.
2 = 283.
8.
The sentence read, "One part was adjusted by adding 270 and the other part remained the same so the whole needed to be adjusted by adding 270." For the second one, we decided to adjust it by subtracting 10,000.
So now we had 24,375 + 4,100 = 28,475.
One part was adjusted by subtracting 10,000 and the other part remained the same so the whole needed to be adjusted by subtracting 10,000.
Let's look at number two then.
We'll start with Izzy who said, "If I adjust one part, then the whole has to be adjusted in the same way to maintain a balanced equation.
This is true for decimal fractions and larger numbers too!" Well, Izzy was correct.
It says, "This is correct.
When one part is adjusted, then the whole needs to be adjusted to ensure the equation is balanced for all types of number." There's an example for decimal fractions.
There's an adjustment of 30 that's been made.
We've got 0.
3 + 0.
4 = 0.
7.
That's adjusted to give 30.
3 + 0.
4 = 30.
7.
Here's Alex's.
"In order to maintain a balanced equation with parts containing decimal fractions, I can only make adjustments by adding or subtracting decimal fractions." This is incorrect.
You can adjust a part using integers even if the numbers feature decimal fractions.
The previous example equation shows this.
There it is again.
And you can see that an integer has been added on here, disproving Alex's theory.
Okay, here's number three then.
We might start by looking at Andeep who had a bonus addition of 1,800.
So he didn't score 7,250, but we didn't need to work out his new score.
We just need to work out the totals.
So that was 22,800.
Sofia had a penalty reduction of 2,500, so her score wasn't that, but again, we didn't need to recalculate her score.
We just needed to recalculate the total, which was 22,500.
So consequently, Sam and Izzy were first.
Jun and Andeep were second.
Aisha and Sofia were third.
Let's summarise what we've learned today.
Equations show two expressions which have equal value.
If one part of the equation is adjusted and the other part remains the same, then the whole needs to be adjusted by the same amount.
This can be used to solve missing digit and number problems as well as worded problems. For efficiency, sometimes the adjusted part doesn't need to be calculated.
Instead, the adjusted whole can be calculated to find the solution.
My name is Mr. Tazzyman.
I've enjoyed learning with you today and I hope you have as well.
I'll see you again soon on another maths lesson.
Bye for now.
