## Envision Math Grade 8, chapter Exercise 1 Standards for Mathematical Practice

**Page 16 Exercise 1 Answer**

Given Nori, her friend, and her mother bought a baseball game ticket package. The package includes good seats, lunch, and a chance to get autographs from players. The total cost for the three of them is $375.

The objective is to find an equation of that shows how much each of them still owes and the other question to consider is have the problem been solved before.

The amount that each still owes can be found by defining a variable to unknown and use the known quantities.

Find the equation of that each owes.

Nori, her friend, and her mother bought a baseball game ticket package.

The package includes good seats, lunch, and a chance to get autographs from players.

The total cost for the three of them is $375.

They each paid a $50 deposit.

Total cost is$375

Each paid $50 deposit.

Define a variable.

x = $50

y = $50

z = $50

Total amount paid as deposit is,

x + y + z = $50 + $50 + $50

= $150

Find the total amount that has to be pay.

Total cost of the ticket is$375

= $375 − $150

=$225

Find the amount that each still owes.

Total amount they owe divided by 3.

= $225 ÷ 3

= $75

The amount that each still owes is $75.

I have solved similar problems before.

The final answer is that each still owes $75 and the similar problem has been solved before.

**Page 16 Exercise 2 Answer**

Given Nori, her friend, and her mother bought a baseball game ticket package. The package includes good seats, lunch, and a chance to get autographs from players. The total cost for the three of them is $375.

The objective is to find that what information are necessary and what are not necessary.

The necessary and unnecessary information can be found by observing the problem.

Find what information are necessary and what are not necessary.

Nori, her friend, and her mother bought a baseball game ticket package.

The package includes good seats, lunch, and a chance to get autographs from players.

The total cost for the three of them is $375.

They each paid a $50 deposit.

Necessary information:

The total cost for the three of them is $375.

They each paid a $50 deposit.

Unnecessary information:

The package includes good seats, lunch, and a chance to get autographs from players

The necessary information is that the total cost for the three of them is$375 and they each paid a $50 deposit.

The unnecessary information is the package includes good seats, lunch, and a chance to get autographs from players.

**Page 16 Exercise 3 Answer**

Given Nori, her friend, and her mother bought a baseball game ticket package. The package includes good seats, lunch, and a chance to get autographs from players. The total cost for the three of them is $375.

The objective is to check whether the answer make sense.

The answer whether it makes sense or not can be found using the results we obtained.

Check whether the answer make sense.

Nori, her friend, and her mother bought a baseball game ticket package. The package includes good seats, lunch, and a chance to get autographs from players. The total cost for the three of them is $375.

The answer found is that the amount that each still owes is $75.

Yes, it makes send because total amount paid is $375.

For each of them the price of the ticket will be $150

Subtract total amount paid minus total deposit amount.

= $375 − $150

= $225

Divide the amount into three,

= $ 225÷3

= $ 75

Yes, the answer makes sense.

**Page 16 Exercise 4 Answer**

The objective is to find whether the solution pathway is same as the classmate.

The amount that each still owes can be found by defining a variable to unknown and use the known quantities.

Find whether the solution pathway is same as the classmate.

Nori, her friend, and her mother bought a baseball game ticket package. The package includes good seats, lunch, and a chance to get autographs from players. The total cost for the three of them is $375.

The solution pathway is,

Total cost is $375

Each paid $50 deposit.

Define a variable.

x = $50

y = $50

z = $50

Total amount paid as deposit is,

x + y + z = $50 + $50 + $50

= $150

Find the total amount that has to be pay.

Total cost of the ticket is $375

= $375 − $150

= $225

Find the amount that each still owes.

Total amount they owe divided by3.

= $225 ÷ 3

= $75

The amount that each still owes is $75.

The solution pathway is same for me as well as for my classmate.

**Page 18 Exercise 2 Answer**

Given Michael’s class is conducting an experiment by tossing a coin. Tails has come up 5 times in a row. That means the next toss will land heads up.

The objective is to find what arguments can be present to defend the conjecture.

The conjecture can be found using the information provided.

Find what arguments can be present to defend the conjecture.

Michael’s class is conducting an experiment by tossing a coin.

The table below shows the results of the last tosses.

The two outcomes of the toss of a coin are heads or tails. For any individual toss of the coin, the outcome will be either heads o. tails. The two outcomes (heads or tails) are therefore mutually exclusive; if the coin comes up heads on a single toss, it cannot come up tails on the same toss.

The arguments that can be present to defend the conjecture is,

A coin can land on heads or tails, so there are two equally likely outcomes.

Because there may be either head or tail.

The arguments that can be present to defend the conjecture is,

A coin can land on heads or tails, so there are two equally likely outcomes.

**Page 18 Exercise 3 Answer**

Given Michael’s class is conducting an experiment by tossing a coin. The table below shows the results of the last 9 tosses.

The objective is to find what conjecture can be made about the solution to the problem.

The conjecture can be found using the information provided.

Find what conjecture can be made about the solution to the problem.

Michael’s class is conducting an experiment by tossing a coin.

The table below shows the results of the last 9 tosses.

The conjecture that can be made about the solution to the problem is,

Total observations=25

Required outcome Tails {T,T,T,T,T}

This can occur only ONCE!

The conjecture made is as there are continuous five tails, this may end up with head at next toss.

**Page 19 Exercise 1 Answer**

Given the class decided to toss two coins at the same time. They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

The objective is to find the representation that is used to show the relationship among quantities or variables.

The representation can be found using the modelling capability.

Find the representation that is used to show the relationship among quantities or variables.

The class decided to toss two coins at the same time.

They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

The possible outcomes are, HH, TT, TH, & HT

Representation of the outcomes is,

The representation of the relationship among the quantities is,

**Page 19 Exercise 2 Answer**

Given the class decided to toss two coins at the same time. They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

The objective is to find the way to make our model to work better.

The representation can be found using the modelling capability.

Find the way to make our model to work better.

The class decided to toss two coins at the same time.

They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more

likely that the coins will show one heads and one tails.

To make the model better if it doesn’t work, use table representation.

If the model doesn’t work, the choose better model like table representation

**Page 19 Exercise 3 Answer**

Given the class decided to toss two coins at the same time. They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

The objective is to find the assumption that can be made to simplify the problem.

The simplified problem can be found using the better assumption.

Determine the assumption that can be made to simplify the problem.

The class decided to toss two coins at the same time.

They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

When we toss two coins simultaneously then the possible of outcomes are:

(Two heads) or (one head and one tail) or (two tails) i.e., in short (H,H)(H,T)(T,T) respectively;

Where H is denoted for head and T is denoted for tail.

The assumption made to simplify the problem is when two coins is tossed simultaneously the possible of outcomes are: (Two heads) or (one head and one tail) or (two tails)

**Page 19 Exercise 4 Answer**

The objective is to check whether the solution make sense.

The simplified problem can be found using the better assumption.

Check whether the solution make sense.

The class decided to toss two coins at the same time. They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

The solution obtained is,

The possible outcomes are,

HH, TT, TH, & HT

The solution makes sense.

Yes. The solution or prediction we had makes sense. Because on flipping two coins the outcomes should be both heads or both tails or one head and one tail.

**Page 19 Exercise 5 Answer**

The objective is to find is there something that is forgotten or not considered.

The representation can be found using the modelling capability.

Find is there something that is forgotten or not considered.

The class decided to toss two coins at the same time. They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

There is nothing that is forgotten or not considered while predicting the solution.

Nothing is forgotten or not considered while predicting the solution.

**Page 20 Exercise 1 Answer**

Given The Golden Company uses signs in the shape of golden rectangles to advertise its products. In a golden rectangle the length of the longer side is about 1.618 times longer than the shorter side. Draw rectangles to scale to create templates of possible small, medium, and large advertising signs.

The objective is to find whether different tools can be used.

The tools that are to be used can be found by strategical thinking.

Find whether different tools can be used.

The Golden Company uses signs in the shape of golden rectangles to advertise its products. In a golden rectangle the length of the longer side is about times longer than the shorter side.

Draw rectangles to scale to create templates of possible small, medium, and large advertising signs.

Large advertising signs.

Medium advertising signs.

Small advertising signs.

Different tools can be used to make design and technology software’s for drawing.

**Page 20 Exercise 2 Answer**

Given The Golden Company uses signs in the shape of golden rectangles to advertise its products. In a golden rectangle the length of the longer side is about times longer than the shorter side. Draw rectangles to scale to create templates of possible small, medium, and large advertising signs.

The objective is to find what other resources can uses to reach the solution

The tools that are to be used can be found by strategical thinking.

Find what other resources can uses to reach the solution.

The Golden Company uses signs in the shape of golden rectangles to advertise its products. In a golden rectangle the length of the longer side is about times longer than the shorter side.

The other resources that can be used to help reach the solution is technical drawing software where the measurements will be given and content to be advertise will also be given.

The technical drawing software is the resource that is used to reach the solution.

**Page 21 Exercise 1 Answer**

Given the explanation about golden rectangle advertisement and their dimensions.

The objective is to state the meaning of variables and symbols used.

The variables and symbols used can be found using the information provided

State the meaning of variables and symbols used.

The work is precise enough and appropriate.

Draw the shorter side and then multiply that dimension by 1.618 to determine the length of the longer side.

Length L = Shorter side x1.618

Width W =1.618 cm

The variables used are,

L for length of the rectangle.

W for width of the rectangle.

Symbols that are used is,

Equal to =

Multiplication x

The symbols used are equal to and multiplication.

The variables used are L and W.

**Page 21 Exercise 2 Answer**

Given the explanation about golden rectangle advertisement and their dimensions.

The objective is to specify the units of measure using.

The variables and symbols used can be found using the information provided.

State the meaning of variables and symbols used.

The work is precise enough and appropriate.

Draw the shorter side and then multiply that dimension by 1.618 to determine the length of the longer side.

Length L = Shorter side x1.618

Width W =1.618cm

The variables used are,

L for length of the rectangle.

W for width of the rectangle.

Symbols that are used is,

Equal to =

Multiplication x

The units of measure that is specified is centimetre or cm.

The units of measure that is specified is centimetre or cm.

**Page 21 Exercise 3 Answer**

Given the explanation about golden rectangle advertisement and their dimensions.

The objective is to find whether the work is precise or exact enough.

The precision used can be found using the result got.

Find whether the work is precise or exact enough.

I am using the appropriate tools to make sure that the dimensions of the templates are precise.

The work is precise and exact because the appropriate tools are used.

**Page 21 Exercise 4 Answer**

Given the explanation about golden rectangle advertisement and their dimensions.

The objective is to find whether the explanation carefully formulated.

The variables and symbols used can be found using the information provided.

Find whether the explanation carefully formulated.

Length L = Shorter side x1.618

Width W = 1.618cm

The variables used are,

L for length of the rectangle.

W for width of the rectangle.

Symbols that are used is,

Equal to =

Multiplication x1

The units of measure that is specified are centimetre or cm.

Yes, the information provided are carefully formulated.

Yes, the explanations provided are carefully formulated.

**Page 22 Exercise 1 Answer**

Given that Stuart is studying cell division. The table below shows the number of cells after a certain number of divisions.

The objective is to make a chart that shows drawings of the cell divisions through 10 divisions.

The chart can be found using the table and the explanation provided.

Make a chart that shows drawings of the cell divisions through 10 divisions.

The table below shows the number of cells after a certain number of divisions.

The structure for the problem is,

1 cell becomes 2 cells and 2 cells become 4 cells, and so on.

The attributes that are used in the drawing is a tree chart to describe the cells and their division.

**Page 22 Exercise 2 Answer**

Given that Stuart is studying cell division. The table below shows the number of cells after a certain number of divisions.

The objective is to find the patterns in numbers that are described.

The patterns in number can be found using observation.

Find the patterns in numbers that are described.

Stuart is studying cell division. The table shows the number of cells after a certain number of divisions.

Initial cell is one and on further cell division, it becomes 2.

2 cells become 4

4 cells become 8 and so on….

1, 2, 4, 8, 16, 32, 64, 128, 256,….

This sequence has a factor of 2 between each number.

Each term (except this first term) is found by multiplying the previous term by 2.

The pattern obtained in a geometric sequence.

The pattern in numbers can be described as GEOMETRIC SEQUENCE.

**Page 22 Exercise 3 Answer**

Given that Stuart is studying cell division. The table below shows the number of cells after a certain number of divisions.

The objective is to find th expressions or equations in different ways.

The patterns in number can be found using observation.

Find the expressions or equations in different ways.

Stuart is studying cell division. The table below shows the number of cells after a certain number of divisions.

1, 2, 4, 8, 16, 32, 64, 128, 256,…..

This sequence has a factor of 2 between each number.

Each term (except the first term) is found by multiplying the previous term by 2.

The pattern obtained in geometric sequence.

The equation of expression that can is noticed is 2n.

Example:

Initial cell is one and on further cell division, it becomes 2.

On second division is should become 4.

For that multiply the cells obtained in initial division by 2.

The expression or equation can be viewed in different ways. The expression we obtained is 2n.

**Page 23 Exercise 1 Answer**

Given the explanation about cell division.

The objective is to find the general method or shortcut to solve the problem.

The general methods can be found using simple formulas and observing.

Find the general method or shortcut to solve the problem.

From the figure, it is observed that the cells are divided from one cell.

The general method that can be used to solve the problem is,

Number of cells multiplied by 2 is the result.

It is noticed that when calculations are repeated.

Then we can find more general methods and shortcuts.

The final answer is that it can be generalized and the shortcut used is power tables as the cells divided is multiplied by two will give the result.

**Page 23 Exercise 2 Answer**

Given the explanation about cell division.

The objective is to find that an expression or equation can be derived.

The general methods can be found using the simple formulas and observation.

Find an expression or equation that can be derived from the examples.

The example obtained is,

The expression or equation that can be derived is,

N = n x 2

Equation:

4 * 2 = 8

Where N = total number of cells.

n = number of cells divided.

The final answer is that an expression that can be derived is n N = n x 2

**Page 23 Exercise 3 Answer**

Given the explanation about cell division.

The objective is to find that how reasonable are the results.

The general methods can be found using the simple formula and observation.

Find that how reasonable are the results.

The expression or equation that can be derived is,

Where N = total number of cells.

n = number of cells divided.

The result obtained is so reasonable that, using expression we obtained, number of cells that are divided can be found out easily without drawing 1024 cells in a chat that takes time hardly.

The final answer is that, the result we are getting is so reasonable that using the formula we obtained, it can be used to solve the cell divide.