Math / Algebra

Question9. $\sqrt{\int}$ Ted thinks that because $10 \cdot 10=100$ and 2 . $5=10$ , he should be able to calculate $12 \cdot 15$ by adding $100+10$ to get 110. Explain to Ted in two different ways that, even though his method is not correct, his calculations can be part of a correct way to calculate $12 \cdot 15$ . by drawing an array by writing equations that use the distributive property

Studdy Solution

STEP 1

What is this asking? We need to show Ted two ways to correctly calculate $12 \cdot 15$ using his initial ideas, but in a way that actually works! Watch out! Ted's initial method isn't correct, so we need to be careful to explain why the correct methods work.

STEP 2

1. Array Method
2. Distributive Property Method

STEP 3

Hey Ted, let's imagine a big rectangle!
This rectangle is 12 units tall and 15 units wide.
We want to find the total number of squares inside, which is the same as calculating $12 \cdot 15$ .

STEP 4

Now, let's break this big rectangle into smaller rectangles using your numbers.
We can split the 12 into 10 and 2, and the 15 into 10 and 5.
This gives us four smaller rectangles!

STEP 5

The first rectangle is $10 \cdot 10 = 100$ squares.
The second rectangle is $2 \cdot 10 = 20$ squares.
The third rectangle is $10 \cdot 5 = 50$ squares.
And the last little rectangle is $2 \cdot 5 = 10$ squares.
See how we used your calculations?

STEP 6

Now, add up the squares in all the smaller rectangles: $100 + 20 + 50 + 10 = 180$ .
So, $12 \cdot 15 = 180$ !

STEP 7

Remember the distributive property?
It says $a \cdot (b + c) = a \cdot b + a \cdot c$ .
We can use this to break down our multiplication.

STEP 8

Let's write $12$ as $10 + 2$ and $15$ as $10 + 5$ .
Now, $12 \cdot 15$ becomes $(10 + 2) \cdot (10 + 5)$ .

STEP 9

Using the distributive property, we get $(10 + 2) \cdot 10 + (10 + 2) \cdot 5$ .

STEP 10

Distribute again!
We have $10 \cdot 10 + 2 \cdot 10 + 10 \cdot 5 + 2 \cdot 5$ .

STEP 11

Look! Your calculations are right there!
Now we just compute: $100 + 20 + 50 + 10 = 180$ .
So, $12 \cdot 15 = 180$ !

STEP 12

$12 \cdot 15 = 180$ .
We showed Ted two ways to use his initial calculations correctly: by visualizing the multiplication as an array and by using the distributive property!

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Studdy Solution

STEP 1

What is this asking? We need to show Ted two ways to correctly calculate $12 \cdot 15$ using his initial ideas, but in a way that actually works! Watch out! Ted's initial method isn't correct, so we need to be careful to explain why the correct methods work.

STEP 2

1. Array Method
2. Distributive Property Method

STEP 3

Hey Ted, let's imagine a big rectangle!
This rectangle is 12 units tall and 15 units wide.
We want to find the total number of squares inside, which is the same as calculating $12 \cdot 15$ .

STEP 4

Now, let's break this big rectangle into smaller rectangles using your numbers.
We can split the 12 into 10 and 2, and the 15 into 10 and 5.
This gives us four smaller rectangles!

STEP 5

The first rectangle is $10 \cdot 10 = 100$ squares.
The second rectangle is $2 \cdot 10 = 20$ squares.
The third rectangle is $10 \cdot 5 = 50$ squares.
And the last little rectangle is $2 \cdot 5 = 10$ squares.
See how we used your calculations?

STEP 6

Now, add up the squares in all the smaller rectangles: $100 + 20 + 50 + 10 = 180$ .
So, $12 \cdot 15 = 180$ !

STEP 7

Remember the distributive property?
It says $a \cdot (b + c) = a \cdot b + a \cdot c$ .
We can use this to break down our multiplication.

STEP 8

Let's write $12$ as $10 + 2$ and $15$ as $10 + 5$ .
Now, $12 \cdot 15$ becomes $(10 + 2) \cdot (10 + 5)$ .

STEP 9

Using the distributive property, we get $(10 + 2) \cdot 10 + (10 + 2) \cdot 5$ .

STEP 10

Distribute again!
We have $10 \cdot 10 + 2 \cdot 10 + 10 \cdot 5 + 2 \cdot 5$ .

STEP 11

Look! Your calculations are right there!
Now we just compute: $100 + 20 + 50 + 10 = 180$ .
So, $12 \cdot 15 = 180$ !

STEP 12

$12 \cdot 15 = 180$ .
We showed Ted two ways to use his initial calculations correctly: by visualizing the multiplication as an array and by using the distributive property!

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Studdy Solution

STEP 1

What is this asking? We need to show Ted two ways to correctly calculate 12⋅1512 \cdot 1512⋅15 using his initial ideas, but in a way that actually works! Watch out! Ted's initial method isn't correct, so we need to be careful to explain *why* the correct methods work.

STEP 2

1. Array Method2. Distributive Property Method

STEP 3

Hey Ted, let's imagine a big rectangle!This rectangle is **12 units** tall and **15 units** wide.We want to find the total number of squares inside, which is the same as calculating 12⋅1512 \cdot 1512⋅15.

STEP 4

Now, let's break this big rectangle into smaller rectangles using your numbers.We can split the **12** into **10** and **2**, and the **15** into **10** and **5**.This gives us four smaller rectangles!

STEP 5

STEP 6

Now, add up the squares in all the smaller rectangles: 100+20+50+10=180100 + 20 + 50 + 10 = 180100+20+50+10=180.So, 12⋅15=18012 \cdot 15 = 18012⋅15=180!

STEP 7

Remember the distributive property?It says a⋅(b+c)=a⋅b+a⋅ca \cdot (b + c) = a \cdot b + a \cdot ca⋅(b+c)=a⋅b+a⋅c.We can use this to break down our multiplication.

STEP 8

Let's write 121212 as 10+210 + 210+2 and 151515 as 10+510 + 510+5.Now, 12⋅1512 \cdot 1512⋅15 becomes (10+2)⋅(10+5)(10 + 2) \cdot (10 + 5)(10+2)⋅(10+5).

STEP 9

Using the distributive property, we get (10+2)⋅10+(10+2)⋅5(10 + 2) \cdot 10 + (10 + 2) \cdot 5(10+2)⋅10+(10+2)⋅5.

STEP 10

Distribute again!We have 10⋅10+2⋅10+10⋅5+2⋅510 \cdot 10 + 2 \cdot 10 + 10 \cdot 5 + 2 \cdot 510⋅10+2⋅10+10⋅5+2⋅5.

STEP 11

Look! Your calculations are right there!Now we just compute: 100+20+50+10=180100 + 20 + 50 + 10 = 180100+20+50+10=180.So, 12⋅15=18012 \cdot 15 = 18012⋅15=180!

STEP 12

12⋅15=18012 \cdot 15 = 18012⋅15=180.We showed Ted two ways to use his initial calculations correctly: by visualizing the multiplication as an array and by using the distributive property!

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to show Ted two ways to correctly calculate $12 \cdot 15$ using his initial ideas, but in a way that actually works! Watch out! Ted's initial method isn't correct, so we need to be careful to explain why the correct methods work.

1. Array Method
2. Distributive Property Method

Hey Ted, let's imagine a big rectangle!
This rectangle is 12 units tall and 15 units wide.
We want to find the total number of squares inside, which is the same as calculating $12 \cdot 15$ .

Now, let's break this big rectangle into smaller rectangles using your numbers.
We can split the 12 into 10 and 2, and the 15 into 10 and 5.
This gives us four smaller rectangles!

Now, add up the squares in all the smaller rectangles: $100 + 20 + 50 + 10 = 180$ .
So, $12 \cdot 15 = 180$ !

Remember the distributive property?
It says $a \cdot (b + c) = a \cdot b + a \cdot c$ .
We can use this to break down our multiplication.

Let's write $12$ as $10 + 2$ and $15$ as $10 + 5$ .
Now, $12 \cdot 15$ becomes $(10 + 2) \cdot (10 + 5)$ .

Using the distributive property, we get $(10 + 2) \cdot 10 + (10 + 2) \cdot 5$ .

Distribute again!
We have $10 \cdot 10 + 2 \cdot 10 + 10 \cdot 5 + 2 \cdot 5$ .

Look! Your calculations are right there!
Now we just compute: $100 + 20 + 50 + 10 = 180$ .
So, $12 \cdot 15 = 180$ !

$12 \cdot 15 = 180$ .
We showed Ted two ways to use his initial calculations correctly: by visualizing the multiplication as an array and by using the distributive property!