Go through the Math in Focus Grade 7 Workbook Answer Key Chapter 1 Lesson 1.3 Introducing Irrational Numbers to finish your assignments.

## Math in Focus Grade 7 Course 2 A Chapter 1 Lesson 1.3 Answer Key Introducing Irrational Numbers

### Math in Focus Grade 7 Chapter 1 Lesson 1.3 Guided Practice Answer Key

**Hands-On Activity**

Materials:

- paper
- ruler
- scissors

Find the value of \(\sqrt{2}\) using a square.

Work in pairs.

Step 1.

Draw a square that has a length of 2 inches on a piece of paper. Then cut out the square.

Step 2.

Find the area of the square (square A).

Step 3.

Fold the four vertices of square A towards the center to form square B as shown below.

Step 4.

State how the areas of square A and square B are related. State the area of square B. How can you represent the length of a side of square B?

Step 5.

Using your answer in step 4, find the length of a side of square B with a calculator. Round your answer to 2 decimal places.

**Math Journal**

Place an edge of square B alongsi& a ruler to measure its length. Explain why the reading from the ruler is different from the answer in step 5.

**Copy and complete.**

Question 1.

Graph \(\sqrt{5}\) on the number line using rational approximations.

Which two whole numbers is \(\sqrt{5}\) between? Justify your reasoning. Using a calculator, \(\sqrt{5}\) = .

Graph an interval where \(\sqrt{5}\) is located.

The value of \(\sqrt{5}\) with two decimal places is

is closer to than to . So, \(\sqrt{5}\) is located closer to .

By using an approximate value of \(\sqrt{5}\), locate \(\sqrt{5}\) on the number line.

Answer:

The \(\sqrt{5}\) is between two whole numbers. The two whole numbers are **2 and 3**.

By using calculator \(\sqrt{5}\) = **2.236067977….**

The value of \(\sqrt{5}\) with two decimal places is **2.24**.

The decimal** 2.24** is closer to **2.2** than to **2.3**.

So, \(\sqrt{5}\) is located closer to **2.2**.

By using an approximate value of \(\sqrt{5}\), located \(\sqrt{5}\) on the number line as we can observe in the below image.

**Copy and complete.**

Question 2.

Graph –\(\sqrt{2}\) on the number line using rational approximations.

Which two integers is –\(\sqrt{2}\) between? Justify your reasoning. Using a calculator, –\(\sqrt{2}\) = .

Graph an interval where –\(\sqrt{2}\) is located.

The value of –\(\sqrt{2}\) with two decimal places is ? .

is closer to than to . So, –\(\sqrt{2}\) is located closer to ?

By using an approximate value of –\(\sqrt{2}\), locate –\(\sqrt{2}\) on the number line.

Answer:

The –\(\sqrt{2}\) is between two whole numbers. The two whole numbers are **-1 and -2**.

By using calculator –\(\sqrt{2}\) = **-1.414213562….**.

The value of –\(\sqrt{2}\) with two decimal places is **-1.41**.

The decimal** -1.41** is closer to **-1.4** than to **-1.5**.

So, –\(\sqrt{2}\) is located closer to **-1.4**.

By using an approximate value of –\(\sqrt{2}\), located –\(\sqrt{2}\) on the number line as we can observe in the below image.

**Solve.**

Question 3.

Graph –\(\sqrt{7}\) on the number line using rational approximations.

Answer:

The –\(\sqrt{7}\) is between two whole numbers. The two whole numbers are **-2 and -3**.

By using calculator –\(\sqrt{7}\) = **-2.645751311….**.

The value of –\(\sqrt{7}\) with two decimal places is **-2.64**.

The decimal** -2.64** is closer to **-2.6** than to **-2.7**.

So, –\(\sqrt{7}\) is located closer to **-2.6**.

By using an approximate value of –\(\sqrt{7}\), located –\(\sqrt{7}\) on the number line as we can observe in the below image.

### Math in Focus Course 2A Practice 1.3 Answer Key

**Locate each positive irrational number on the number line using rational approximations. First tell which two whole numbers the square root is between.**

Question 1.

\(\sqrt{3}\)

Answer:

The \(\sqrt{3}\) is between two whole numbers. The two whole numbers are **1 and 2**.

By using calculator \(\sqrt{3}\) = **1.732050807….**

The value of \(\sqrt{3}\) with two decimal places is **1.73**.

The decimal** 1.73** is closer to **1.7** than to **1.8**.

So, \(\sqrt{3}\) is located closer to **1.7**.

By using an approximate value of \(\sqrt{3}\), the positive irrational number \(\sqrt{3}\) is located on the number line as we can observe in the above image.

Question 2.

\(\sqrt{7}\)

Answer:

The \(\sqrt{7}\) is between two whole numbers. The two whole numbers are **2 and 3**.

By using calculator \(\sqrt{7}\) = **2.645751311….**

The value of \(\sqrt{7}\) with two decimal places is **2.64**.

The decimal** 2.64** is closer to **2.6** than to **2.7**.

So, \(\sqrt{7}\) is located closer to **2.6**.

By using an approximate value of \(\sqrt{7}\), the positive irrational number \(\sqrt{7}\) is located on the number line as we can observe in the above image.

Question 3.

\(\sqrt{11}\)

Answer:

The \(\sqrt{11}\) is between two whole numbers. The two whole numbers are **3 and 4**.

By using calculator \(\sqrt{11}\) = **3.316624790….**

The value of \(\sqrt{11}\) with two decimal places is **3.31**.

The decimal** 3.31** is closer to **3.3** than to **3.4**.

So, \(\sqrt{11}\) is located closer to **3.3**.

By using an approximate value of \(\sqrt{11}\), the positive irrational number \(\sqrt{11}\) is located on the number line as we can observe in the above image.

Question 4.

\(\sqrt{26}\)

Answer:

The \(\sqrt{26}\) is between two whole numbers. The two whole numbers are **5 and 6**.

By using calculator \(\sqrt{26}\) = **5.099019513….**

The value of \(\sqrt{26}\) with two decimal places is **5.09**.

The decimal** 5.09** is closer to **5.1** than to **5.0**.

So, \(\sqrt{26}\) is located closer to **5.1**.

By using an approximate value of \(\sqrt{26}\), the positive irrational number \(\sqrt{26}\) is located on the number line as we can observe in the above image.

Question 5.

\(\sqrt{34}\)

Answer:

The \(\sqrt{34}\) is between two whole numbers. The two whole numbers are **5 and 6**.

By using calculator \(\sqrt{34}\) = **5.830951894….**

The value of \(\sqrt{34}\) with two decimal places is **5.83**.

The decimal** 5.83** is closer to **5.8** than to **5.9**.

So, \(\sqrt{34}\) is located closer to **5.8**.

By using an approximate value of \(\sqrt{34}\), the positive irrational number \(\sqrt{34}\) is located on the number line as we can observe in the above image.

Question 6.

\(\sqrt{48}\)

Answer:

The \(\sqrt{48}\) is between two whole numbers. The two whole numbers are **6 and 7**.

By using calculator \(\sqrt{48}\) = **6.928203230….**

The value of \(\sqrt{48}\) with two decimal places is **6.92**.

The decimal** 6.92** is closer to **6.9** than to **7.0**.

So, \(\sqrt{48}\) is located closer to **6.9**.

By using an approximate value of \(\sqrt{48}\), the positive irrational number \(\sqrt{48}\) is located on the number line as we can observe in the above image.

**Locate each negative irrational number on the number line using rational approximations. First tell which two integers the square root is between.**

Question 7.

–\(\sqrt{5}\)

Answer:

The –\(\sqrt{5}\) is between two integers. The two integers are **-2 and -3**.

By using calculator –\(\sqrt{5}\) = **-2.236067977….**.

The value of –\(\sqrt{5}\) with two decimal places is **-2.23**.

The decimal** -2.23** is closer to **-2.2** than to **-2.3**.

So, –\(\sqrt{5}\) is located closer to **-2.2**.

By using an approximate value of –\(\sqrt{5}\), the negative irrational number –\(\sqrt{5}\) is located on the number line as we can observe in the above image.

Question 8.

–\(\sqrt{6}\)

Answer:

The –\(\sqrt{6}\) is between two integers. The two integers are **-2 and -3**.

By using calculator –\(\sqrt{6}\) = **-2.449489742….**.

The value of –\(\sqrt{6}\) with two decimal places is **-2.44**.

The decimal** -2.44** is closer to **-2.4** than to **-2.5**.

So, –\(\sqrt{6}\) is located closer to **-2.4**.

By using an approximate value of –\(\sqrt{6}\), the negative irrational number –\(\sqrt{6}\) is located on the number line as we can observe in the above image.

Question 9.

–\(\sqrt{17}\)

Answer:

The –\(\sqrt{17}\) is between two integers. The two integers are **-4 and -5**.

By using calculator –\(\sqrt{17}\) = **-4.123105625….**.

The value of –\(\sqrt{17}\) with two decimal places is **-4.12**.

The decimal** -4.12** is closer to **-4.1** than to **-4.2**.

So, –\(\sqrt{17}\) is located closer to **-4.1**.

By using an approximate value of –\(\sqrt{17}\), the negative irrational number –\(\sqrt{17}\) is located on the number line as we can observe in the above image.

Question 10.

–\(\sqrt{26}\)

Answer:

The –\(\sqrt{26}\) is between two integers. The two integers are **-5 and -6**.

By using calculator –\(\sqrt{26}\) = **-5.099019513…**.

The value of –\(\sqrt{26}\) with two decimal places is **-5.09**.

The decimal** -5.09** is closer to **-5.1** than to **-5.0**.

So, –\(\sqrt{26}\) is located closer to **-5.1**.

By using an approximate value of –\(\sqrt{26}\), the negative irrational number –\(\sqrt{26}\) is located on the number line as we can observe in the above image.

Question 11.

–\(\sqrt{53}\)

Answer:

The –\(\sqrt{53}\) is between two integers. The two integers are **-7 and -8**.

By using calculator –\(\sqrt{53}\) = **-7.280109889…**.

The value of –\(\sqrt{53}\) with two decimal places is **-7.28**.

The decimal** -7.28** is closer to **-7.3** than to **-7.2**.

So, –\(\sqrt{53}\) is located closer to **-7.3**.

By using an approximate value of –\(\sqrt{53}\), the negative irrational number –\(\sqrt{53}\) is located on the number line as we can observe in the above image.

Question 12.

–\(\sqrt{80}\)

Answer:

The –\(\sqrt{80}\) is between two integers. The two integers are **-8 and -9**.

By using calculator –\(\sqrt{80}\) = **-8.944271909…**.

The value of –\(\sqrt{80}\) with two decimal places is **-8.94**.

The decimal** -8.94** is closer to **-8.9** than to **-9.0**.

So, –\(\sqrt{80}\) is located closer to **-8.9**.

By using an approximate value of –\(\sqrt{80}\), the negative irrational number –\(\sqrt{80}\) is located on the number line as we can observe in the above image.

**Use a calculator. Locate each irrational number to 3 decimal places on the number line using rational approximations.**

Question 13.

\(\sqrt{47}\)

Answer:

By using calculator \(\sqrt{47}\) = **6.855654600….**

The value of \(\sqrt{47}\) with three decimal places is **6.855**.

The decimal** 6.855** is closer to **6.86** than to **6.85**.

So, \(\sqrt{47}\) is located closer to **6.86**.

The given irrational number \(\sqrt{47}\) is located on the number line by using rational approximation as we can observe in the above image.

Question 14.

–\(\sqrt{15}\)

Answer:

By using calculator –\(\sqrt{15}\) = –**3.872983346….**

The value of –\(\sqrt{15}\) with three decimal places is –**3.872**.

The decimal** -3.872 **is closer to –**3.87** than to –**3.88**.

So, –\(\sqrt{15}\) is located closer to **-3.87**.

The given irrational number –\(\sqrt{15}\) is located on the number line by using rational approximation as we can observe in the above image.

Question 15.

Answer:

By using calculator = **4.54683594….**

The value of with three decimal places is **4.546**.

The decimal **4.546 **is in between **4.5** and **4.6**.

So, is located on **4.55**.

The given irrational number is located on the number line by using rational approximation as we can observe in the above image.

**Locate each irrational number on the number line using rational approximations.**

Question 16.

\(\sqrt{101}\)

Answer:

By using calculator \(\sqrt{101}\) = **10.049875….**

The value of \(\sqrt{101}\) with two decimal places is **10.04**.

The decimal** 10.04** is closer to **10** than to **10.1**.

So, \(\sqrt{101}\) is located closer to **10**.

The given irrational number \(\sqrt{101}\) is located on the number line by using rational approximation as we can observe in the above image.

Question 17.

–\(\sqrt{132}\)

Answer:

By using calculator –\(\sqrt{132}\) = – **11.489125….**

The value of –\(\sqrt{132}\) with two decimal places is –**11.48**.

The decimal** -11.48 **is closer to –**11.5** than to –**11.4**.

So, –\(\sqrt{132}\) is located closer to –**11.5**.

The given irrational number –\(\sqrt{132}\) is located on the number line by using rational approximation as we can observe in the above image.

Question 18.

\(\sqrt{2,255}\)

Answer:

By using calculator \(\sqrt{2,255}\) = **47.4868….**

The value of \(\sqrt{2,2551}\) with two decimal places is **47.48**.

The decimal** 47.48** is closer to **47.5** than to **47.4**.

So, \(\sqrt{2,255}\) is located closer to **47.5**.

The given irrational number \(\sqrt{2,255}\) is located on the number line by using rational approximation as we can observe in the above image.

**Solve.**

Question 19.

Locate the value of the constant, π, on the number line using rational numbers.

Answer:

We know that π = 3.14159265

In the above image we can observe the value of the constant, π, is located on the number line using rational numbers.

Question 20.

3.1416 and \(\frac{22}{7}\) are two rational approximate values of π.

a) Graph 3.1416, \(\frac{22}{7}\), and π on the number line.

b) Which of the two rational approximate values is closer to π?

Answer:

a)We know that 22/7 = 3.1428, π = 3.14159

In the above image we can observe 3.1416, 22/7 and π on the number line.

b) The two rational approximate values closer to π are 3.1416 and 22/7.

Question 21.

A triangle is cut from a square as shown in the diagram. The area of the square is 59 square inches. Approximate the height of the triangle to 3 decimal places.

Answer:

Question 22.

**Math Journal** When do you need to approximate an irrational number with a rational value? Explain and illustrate with an example.

Answer: