Big Ideas Math Integrated Math 1 Student Journal Solutions Chapter 9 Maintaining Mathematical Proficiency Exercise

Big Ideas Math Integrated Math Chapter 9 Maintaining Mathematical Proficiency

 

246  Exercise 1  Problem 1

Question 1.

Given the arithmetic sequence 22, 34, 46, 58, …

1. Derive the formula for the nth term of the sequence.
2. Use the formula to find the 20th term of the sequence.

Answer:

Given

The arithmetic sequence 22, 34, 46, 58, …

To find the nth term and 20th term of the sequence 22,34,46,58 …………..

Here, the sequence is  22,34,46,58 ………..

​⇒ a = 22

⇒  d = 34 − 22

= 12
​
The nth term of the sequence is a_n

=  a + (n−1)d

​⇒  a_n =  22 +(n−1)12

⇒ a_n = 22 + 12n−12

⇒ a_n = 12n + 10

​The 20^th  term is a₂₀

= 12(20) + 10

​a₂₀ = 240 + 10

a₂₀ = 250

​The n^th term of the sequence is  a_n= 12n + 10  and  a₂₀ = 250.

 

Page 246  Exercise 2  Problem 2

Question 2.

Given the arithmetic sequence −13, 0, 13, 26, …

1. Derive the formula for the nth term of the sequence.
2. Use the formula to find the 20th term of the sequence.

Answer:

Given

The arithmetic sequence −13, 0, 13, 26, …

To find the n^th term and a₂₀

Here, the sequence is −13,0,13,26….

​⇒  a = −13

⇒  d = 0−(−13)

= 13

d = 13

​The nth term of the sequence is :

a_n  = a + (n−1)d

​⇒  a_n  = −13 + (n−1)13

⇒  a_n = −13 + 13n − 13

⇒  a_n  = 13n− 26
​
The ​20^th term is a₂₀

=  13(20) − 26

​⇒  260 − 26

⇒  234
​
The nth term  of the sequence  is  a_n=  13n−26  and  a₂₀ = 234

 

Page 246   Exercise 3  Problem 3

Question 3.

Given the arithmetic sequence −4.5, −4.0, −3.5, −3.0, …

1. Derive the formula for the nth term of the sequence.
2. Use the formula to find the 20th term of the sequence.

Answer:

Given

The arithmetic sequence −4.5, −4.0, −3.5, −3.0, …

To find n^th term and a₂₀

Here, the sequence is −4.5,−4.0,−3.5,−3.0, ………..

a = − 4.5

​d = −4.0 − (−4.5)

d = 0.5
​
The n^th  term of the sequence an

=  a + (n−1)d

​⇒  a_n =−4.5 + (n−1)(0.5)

⇒  a_n =−4.5 + 0.5n − 0.5

⇒  a_n = 0.5n − 5.0
​
The 20^th term of the sequence is a₂₀

=  0.5(20) − 5.0

​⇒  a₂₀ = 10.0 − 5.0

⇒  a₂₀ = 5.0
​
The nth ^th of the sequence is a_n = 0.5n − 5.0 and  a₂₀ = 5.0

 

Page 246  Exercise 4 Problem 4

Question 4.

Given the arithmetic sequence −12, 12, 32, 52, …

1. Derive the formula for the nth term of the sequence.
2. Use the formula to find the 20th term of the sequence.

Answer:

Given

The arithmetic sequence −12, 12, 32, 52, …

To find the n^th term of the sequence and a₂₀ term of the sequence.

Here, the sequence is \(\frac{-1}{2}, \frac{1}{2}, \frac{3}{2}, \frac{5}{2}\)

a = \(\frac{−1}{2}\)

d = \(\frac{1}{2}\) − (\(\frac{−1}{2}\))

d = 1

The n^th term of the sequence is an = a + (n−1)d

​⇒  a_n = \(\frac{1}{2}\)+ (n−1)1

⇒  a_n = \(\frac{1}{2}\) + n − 1

⇒  a_n  = n−\(\frac{3}{2}\)

The 20^th term of the sequence is a₂₀

= 20− \(\frac{3}{2}\)

⇒  \(\frac{37}{2}\)

The n^th  term of the sequence is a_n = n− \(\frac{3}{2}\) and  a₂₀ = \(\frac{37}{2}\)

 

Page 246  Exercise 5  Problem 5

Question 5.

Solve for x in the equation: 3x − 9 = 12.

Answer:

Given

Solve for x: 3x − 9 = 12

The equation is  3x − 9 = 12

​⇒  3x = 12 + 9

⇒  3x = 21

⇒  x = 7
​
The value for x = 7

 

Page 246  Exercise 6  Problem 6

Question 6.

Solve for y in the equation: 16 − 4y = 40.

Answer:

Given:  16−4y = 40

To find –  The value of y.

Calculate the value y

The equation is 16 − 4y = 40

⇒ −4y = 40 − 16

⇒ −4y = 4

⇒  y = −6

The solution of the equation 16−4y = 40 is y = −6.

 

Page 246  Exercise 7  Problem 7

Question 7.

Solve for y in the equation: 6z + 4 = 23.

Answer:

Given: 6z + 5 = 23

Put all the terms without z

On the right side to find the value of z.

Given the equation:

6z + 5 = 23

⇒ 6z = 23

⇒  z =  \(\frac{18}{6}\)

⇒  z = 3

The value of z = 3.

 

Page 246  Exercise 8  Problem 8

Question 8.

Solve for q in the equation: 15 = 11q – 23.

Answer:

Given: 15 = 11q − 18.

Put all the terms without q.

On the right side to find the value of q

Given the equation:

​15 = 11q − 18

−11q = −18 − 15

−11q = −33

11q = 33

q = \(\frac{33}{11}\)

q = 3
​
The value of q = 3.

 

Page 246   Exercise 9  Problem 9

Question 9.

Solve for r in the equation: 6r + 3 = 33.

Answer:

Given:  6r + 3 = 33.

Put all the terms without r.

On the right side to find the value of r.

Given the equation:

​6r + 3 = 33

6r = 33 − 3

6r = 30

r = \(\frac{30}{6}\)

r = 5
​
The value of r = 5.

 

Page 246  Exercise 10  Problem 10

Question 10.

Solve for s in the equation: 27 = 4s – 9.

Answer:

Given: 27 = 4s−9.

Put all the terms without s.

On the right side to find the value of s.

Given the equation:

27 = 4s − 9

−4s = −9 − 27

−4s = −36

4s = 36

s = \(\frac{36}{4}\)

s = 9

The value of s = 9.

 

Page 247  Exercise 11  Problem 11

Question 11.

Consider the conditional statement: “If a student studies hard, then the student will pass the exam.”

1. Identify the hypothesis and the conclusion in the given conditional statement.
2. Using symbols, represent the conditional statement.
3. Construct a truth table for the conditional statement p → q, where p represents “the student studies hard” and q represents “the student will pass the exam”.
4. Explain under what condition the conditional statement p → q is false.

Answer:

When a conditional statement is written in if-then form, the “if” part contains the hypothesis and the “then” part contains the conclusion.

Words If p, then q. Symbols p → q (read as “p implies q”)

A conditional statement is symbolized by

p → q it is an if-then statement in which p is a hypothesis and q is conclusion. The logical connector in a conditional statement is denoted by the symbol →.

The conditional is defined to be true unless a true hypothesis leads to a false conclusion.

A truth table for p → q is shown below.

A conditional statement is only false when the hypothesis (p) is true and the conclusion (q) is false, otherwise it is true.

 

Page 248  Exercise 12  Problem 12

Question 12.

Given the points A = (3,0), B = (4,0), and D = (0,0):

1. Show that ADC is a right triangle by identifying the right angle.
2. Calculate the lengths of DA and DC using the distance formula.
3. Using the Pythagorean Theorem, verify that AC is the hypotenuse of the right triangle ΔADC.
4. Prove that the Pythagorean Theorem holds for ΔADC by showing that AC² = AD² + DC².

Answer:

Every right angle satisfy the Pythagoras theorem ΔADC is an right angle triangle.

So, it holds Pythagoras theorem.

ΔADC is an right angle triangle

A = (3,0) , B = (4,0) , D = (0,0)

DA = 3

DC = 4

Therefore by distance formula

AC = \(\sqrt{(4-0)^2+(0-3)^2}\)

AC = \(\sqrt{16+9}\)

AC = 5

Now 5² = 3² + 4²

Therefore, AC² =  AD² + DC²

It is the statement of “Pythagoras” theorem

It implies ΔADC holds Pythagoras theorem.

 

The given statement “If ΔADC  is a right triangle, then the Pythagorean Theorem is valid for ΔADC ” is true.

 

Page 248  Exercise 12  Problem 13

Question 13.

Consider the statement: ” If ∠A and ∠B are complementary, then the sum of their measures is 180°.”

1. Define what it means for two angles to be complementary.
2. Using the definition, explain why the given statement is false.
3. Provide a correct statement about the sum of the measures of complementary angles.
4. Given that ∠A = 35°, find the measure of ∠B if A and ∠B are complementary.

Answer:

We will use the definition of complementary angle to show that given statement is false.

∠A and ∠B are complementary.

Therefore, ∠A+∠B = 90°

It implies sum of their angle measures is not equal to 180°

The given statement ” If ∠A and ∠B are complementary, then the sum of their measures is 180° “is false.

 

Page 248  Exercise 12  Problem 14

Question 14.

Consider the statement: “If figure ABCD is a quadrilateral, then the sum of its angle measures is 180°.”

1. Explain why the given statement is false.
2. State the correct sum of the interior angles of a quadrilateral.
3. Using the angle sum property of a quadrilateral, calculate the sum of the interior angles.
4. Verify your result by using the formula 180° ×(n−2), where n is the number of sides in the quadrilateral.

Answer:

Given statement is

If figure ABCD is a quadrilateral, then the sum of its angle measures is 180°

We will use the angle sum property of a quadrilateral

A quadrilateral has four sides. sum of their angle measure is 180∘ (n−2) where n is number of sides.

A Quadrilateral has four sides.

Therefore, sum of their angles = 180°(n−2) where n is number of sides

= 180° (4−2) 

= 360°

So, sum of their angle measure is 360°

Given statement ” if   ABCD is a quadrilateral, then the sum of its angle measures is 180°” is false.

 

Page 248  Exercise 12  Problem 15

Question 15.

Consider the statement: “If points A, B, and C are collinear, then they lie on the same line.”

1. Explain why the given statement is true.
2. Using the slope property of a line, demonstrate how to check if points A, B, and C are collinear.
3. Given the coordinates of points A(1,2), B(3,6), and C(5,10), use the slope property to verify if these points are collinear.

Answer:

Given statement is

If points A, B, and C are “Collinear”, then they lie on the same line.

Using the slope property of a line we will check whether the points are colinear.

As point A , B , C are Collinear

Slope of AB =

Slope of BC

= Slope of AC

= m

It implies they all lie on the same line.

The given statement If points A, B, and C are collinear, then they lie on the same line” is true.

 

Page 248  Exercise 13  Problem 16

Question 16.

Consider the definition of a conditional statement: “A conditional statement holds true when the result of the first statement implies the other statement. In other words, they either have the same meaning or are properties.”

1. Explain when a conditional statement is true.
2. Explain when a conditional statement is false.
3. Provide an example of a true conditional statement and justify why it is true.
4. Provide an example of a false conditional statement and justify why it is false.

Answer:

A conditional statement holds true when the result of the first statement implies the other statement.

In other words, they either have the same meaning or are properties.

To show conditional statement to be true.

We need to satisfy both statements with each other.

Conditional statement will be false when first statement does not imply second.

A conditional statement can be true or false.

 

Page 248  Exercise 14  Problem 17

Question 17.

Consider the following conditional statements:

1. “If a rectangle does not have 4 sides, then a square is not a quadrilateral.”
2. “If 9 is composite, then 8 is an odd number.”
3. Identify the hypothesis and conclusion in each conditional statement.
4. Explain why the first conditional statement is true.
5. Explain why the second conditional statement is false.
6. Provide an additional example of a true conditional statement and justify why it is true.
7. Provide an additional example of a false conditional statement and justify why it is false.

Answer:

True conditional statement:

If a rectangle does not have 4 sides, then a square is not a quadrilateral.

Hypothesis- A rectangle does not have 4 sides.

Conclusion- A square is not a quadrilateral.

∵   Both hypothesis and conclusion statement are false so the conditional statement is true.

False conditional statement:

If 9 is composite, then 8 is an odd number.

Hypothesis- 9 is composite.

Conclusion- 8 is an odd number.

∵  Hypothesis is true and conclusion is false so the conditional statement is false.

The final answer is that the: True conditional statement is “If a rectangle does not have 4 sides, then a square is not a quadrilateral. “False conditional statement is “If 9 is composite, then 8 is an odd number.”

 

Page 252  Exercise 15  Problem 18

Question 18.

Given the conditional statement: “13x – 5 = -18, because x = -1.”

1. Rewrite the given conditional statement in if-then form.
2. Solve the equation 13x − 5 = −18 to verify the value of x.

Answer:

Given: A conditional statement 13x−5 = −18 , because x = −1.

To find – Rewrite the given conditional statement in if-then form.

Required if – Then form of given conditional statement is

“If 13x−5 = 18 , then x = −1.”

The final answer is that the required if-then statement of given conditional statement is“If 13x−5 = −18 , then x = −1 .”

 

Page 252  Exercise 16  Problem 19

Question 19.

Given the conditional statement: “The sum of the measures of interior angles of a triangle is 180°.”

1. Rewrite the given conditional statement in if-then form.
2. Explain why the rewritten statement is logically equivalent to the original statement.

Answer:

Given: A conditional statement “The sum of the measures of interior angles of a triangle is 180°.”

To find – Rewrite the given conditional statement in if-then form.

Required if- Then the form of given conditional statement is-

“If a shape is a triangle, then the sum of the measures of its interior angles is 180°.”

The final answer is that the required if-then statement of given conditional statement is “If a shape is a triangle, then the sum of the measures of its interior angles is 180°.”

 

Page 252 Exercise 17  Problem 20

Question 20.

A diagram and the statement “LM bisects JK” were given.

1. Let p be the statement “Given diagram” and q be the statement “LM bisects JK.”
2. Explain how you can determine from the diagram that LM bisects JK.
3. Verify whether the given statement q is true based on the diagram.
4. If the diagram shows that JR bisects RK, explain how this information supports the truth of the statement that LM bisects JK.

Answer:

Given: A diagram and a statement \(\overline{L M}\) bisects \(\overline{J K}\).

To find – Whether the given statement is true about the given diagram

Let p  Given diagram and q  \(\overline{L M}\) bisects \(\overline{J K}\)are two statements.

From the diagram we can say that \(\overline{L M}\)  bisects \(\overline{J K}\) because \(\overline{J R}\) bisects \(\overline{R K}\)

⇒  If the diagram is true then \(\overline{J K}\) because \(\overline{J R}\) bisects \(\overline{R K}\).

⇒ The given statement is true about the given diagram.

The final answer is that the given statement  \(\overline{J R}\)  bisects \(\overline{R K}\)  is true about the given diagram.

 

Page 252  Exercise 18  Problem 21

Question 21.

Given a diagram and the statement “∠JRP and ∠PRL are complementary.”

1. Let p be the statement “Given diagram” and q be the statement “∠JRP and ∠PRL are complementary.”
2. Use the diagram to verify whether ∠JRP + ∠PRL = 90°.
3. Explain why the angles ∠JRP and ∠PRL are complementary based on the diagram.
4. Conclude whether the given statement q is true about the diagram.

Answer:

Given: A diagram and a statement ∠JRP and ∠PRL are complementary.

To find – whether the given statement is true about the given diagram.

Let p  Given diagram  and q ∠JRP and ∠PRL are complementary” are two statements

∵ ∠JRP + ∠PRL = 90° (From the diagram)

⇒   ∠JRP and ∠PRL are complementary angles.

⇒   If the given diagram is true then∠JRP  and ∠PRL are complementary angles.

⇒  The given statement is true about the given diagram.

The final answer is that the given statement ∠JRP and ∠PRL are complementary angles is true about the given diagram.

 

Page 252  Exercise 19  Problem 22

Question 22.

A diagram and the statement “∠MRQ ≅ ∠PRL” were given.

1. Let p be the statement “Given diagram” and q be the statement “∠MRQ ≅ ∠PRL.”
2. Explain the relationship between ∠MRQ and ∠PRL using the properties of the diagram.
3. Verify whether ∠MRQ and ∠PRL are congruent based on the diagram.
4. Conclude whether the given statement q is true about the diagram.

Answer:

Given: A diagram and a statement ∠MRQ ≅ ∠PRL.

To find – whether the given statement is true about the given diagram.

Let p :  Given diagram and q  ∠MRQ ≅ ∠PRL  are two statements.

∵ ∠MRQ and ∠PRL are vertically opposite angles.

⇒ ∠MRQ ≅ ∠PRL.

⇒  If the given diagram is true then∠MRQ ≅ ∠PRL.

⇒  The given statement is true about the given diagram.

The final answer is that the given statement  ∠MRQ ≅ ∠PRL is true about the given diagram.

 

Page 253  Exercise 20  Problem 23

Question 23.

Consider the various uses of reasoning to solve problems as described below:

1. Reasoning is used to find a property that a series of steps shows.
2. Reasoning is used to find a general rule for a number of specific observations.
3. Reasoning is used so that factual statements can come to a logical conclusion.
4. Reasoning is also used to find a pattern in a series of observations.
5. Explain how reasoning can be used to find a property that a series of steps shows. Provide an example.
6. Describe how reasoning can help find a general rule from specific observations. Provide an example.
7. Explain how reasoning can lead factual statements to a logical conclusion. Provide an example.
8. Describe how reasoning is used to find patterns in a series of observations. Provide an example.

Answer:

Uses of reasoning to solve problems:

Reasoning is used to find a property that a series of steps shows.

Reasoning is used to find a general rule for a number of specific observations.

Reasoning is used to factual statements can come to a logical conclusion.

Reasoning is also used to find a pattern in a series of observations.

The final answer is that the uses of reasoning to solve a problem is:

Reasoning is used to find a property that a series of steps shows.

Reasoning is used to find a general rule for a number of specific observations.

Reasoning is used to factual statements can come to a logical conclusion.

Reasoning is also used to find a pattern in a series of observations.

 

Page 253  Exercise 21  Problem 24

Question 24.

Given a pattern shown in a diagram series:

1. Identify and describe the pattern in the sequence of diagrams.
2. Make a conjecture about the pattern based on your observations.
3. Determine which diagram will be the 10th object in the sequence.
4. Draw the 10th object based on your conjecture.

Answer:

Given: A pattern shown in a Diagram.

To find –  A conjecture about the pattern and draw the 10th object.

We can see that the Diagrams included here are repeating after every three Diagrams.

First Diagram is

Then second Diagram is

And the third Diagram is

And it keeps on repeating.

⇒  The conjecture about the pattern is that the circle is rotating from one vertex of triangle to the next vertex in clockwise direction.

⇒  10th object will be the 1st object only as difference between 10 and 1 is 9 which is a multiple of 3.

⇒  10th object is

The final answer is that the conjecture about the pattern is that the circle is rotating from one vertex of triangle to the next vertex in clockwise direction. and its 10th object is

 

Page 253  Exercise 21  Problem 25

Question 25.

Draw a pattern shown in a series of coordinate plane diagrams:

1. Identify and describe the pattern in the sequence of diagrams.
2. Make a conjecture about the pattern based on your observations.
3. Determine which diagram will be the 10th object in the sequence.
4. Draw the 10th object based on your conjecture.

Answer:

Given:  A pattern shown in a Diagram.

To find –  A conjecture about the pattern and draw the 10th object.

We can see that the Diagrams included here shows a pattern.

All the Diagrams include a 2D coordinate plane.

In first Diagram, there is a curve which is concave down in 1st quadrant.

In second Diagram, there is a straight line in 4th quadrant.

In third Diagram, there is a curve which is concave down in 3rd quadrant.

In fourth Diagram, there is a straight line in 2nd quadrant.

⇒  At even places there are straight lines in even quadrant (first 4th and then 2nd ).

At odd places there are curves which are concave up and concave down in odd quadrant (first two concave down and then two concave up).

⇒ The conjecture about the pattern is that there is a curve in odd quadrant that is repeating after every four Diagram and there is a line in even quadrant that is repeating after every two Diagram .

Place will be a line and 10 is 5th odd number so it will be same as 2nd object.

⇒  Object at the 10th place is

The final answer is that the conjecture about the pattern is that there is a curve in odd quadrant that is repeating after every four Diagrams and there is a line in even quadrant that is repeating after every two  digram  and the 10th object is

 

 Page 254  Exercise 21  Problem 26

Question 26.

Draw a pattern shown in a series of diagrams:

1. Identify and describe the pattern in the sequence of diagrams.
2. Make a conjecture about the pattern based on your observations.
3. Determine which diagram will be the 10th object in the sequence.
4. Draw the 10th object based on your conjecture.

Answer:

Given:  A pattern shown in a diagram.

 

To find – A conjecture about the pattern and draw the 10th object.

We can see from the series of diagram that there is an arrangement in the first three diagram and next three are the mirror images of the first three and then it is repeating as the first six images.

⇒  The conjecture about the pattern is that there are first three arrangements and next three are mirror images of first three and then it is repeating as first six arrangements.

⇒  10th object will be same as the 4th object which is mirror image of 1st object.

⇒  10th object is

The final answer is that the conjecture about the pattern is that there are first three arrangements and next three are mirror images of first three and then it is repeating as first six arrangements.

 

Page 254   Exercise 22  Problem 27 

Question 27.

Provide an example of how you used reasoning to solve a real-life problem. Use the following scenario as a basis:

You want to go for a drive outside, but it is freezing out.

1. Describe the initial situation and any relevant facts or observations.
2. Analyze the situation using reasoning to identify potential risks or concerns.
3. Formulate a conclusion based on your analysis.
4. Support your conclusion with additional evidence or reasoning.

Answer:

Here it is asked to give an example of how you used reasoning to solve a real-life problem.

Suppose that the current situation is that you want to go for a drive outside and it is freezing out.

So, first, we need to analyze the situation, that is for your safety on road the fact that freezing is to be taken care of.

So we can form a conclusion that since it’s freezing it will be dangerous on the road.

Finally, we can support our conclusion by looking into the number of accidents on the road due to freezing.

Therefore, an example of how to use reasoning to solve a real-life problem is shown.