Unit 1: THE Number system U1...Cube Towers Task Activity 1: With the cubes, build the smallest cube tower. The length of the sides of the tower should measure 1 unit. To find the volume

PRE-ALGEBRAUNIT 1:

NUMBER SENSE

Rational Numbers

http://www.shmoop.com/video/rational-irrational-numbers

UNIT 1: NUMBER SENSE

Learning Objective Target 1:

I am learning to convert a decimal expansion that repeats eventually into a rational number.

Rational NumbersGuided Notes

A rational number is any number that can be written as a ratio in the form 𝒂

𝒃where a

and b are integers and b is not equal to 0. What is a ratio? What is integer?

Examples of rational numbers would be 6 (because it can be written as 𝟔

𝟏)

and 0.5 (because it can be written as 𝟏

𝟐)

Every rational number can be written as a terminating decimal or a repeatingdecimal.

A terminating decimal, such as 0.5, has a finite number of decimals. What does finite mean?

A repeating decimal , such as 0.𝟖, has a block of one or more digits that repeat infinitely. What does infinitely mean?

Rational Numbers Journal Entry

Example 1: Consider the given image:

What are some ways we can express the amount of pizza?

Are these values examples of rational numbers?

Rational Numbers


Talk Partners: Consider the given image:

What are some ways we can express the amount of pizza?


Rational Numbers


Example 2: Consider the following image:

What are some ways we can express the value shown?


Rational Numbers


Talk Partners: Consider the following image:

What are some ways we can express the value shown?


Rational Numbers

Express the value shown as a fraction, decimal andpercent.

Are these values you expressed examples of rationalnumbers? Explain your reasoning.

Can you write an equivalent fraction for your givenfraction?

Please use your colored dots to show your confidence in your answer :

Pink: I have no idea how to do this problem so I made my best guessOrange: I think I knew how to do some parts of the problem but feel unsure.Green: I really understood this problem and am confident my answer is correct.

Formative Journal EntryColored Dots

Consider the following image:

Writing Repeating Decimals in Rational FormJournal Entry

Example 1: Consider the following number: 0.𝟑 What are some other ways we can express this value?


0.𝟑


Example 2: Consider the following number: 0.𝟕𝟐 What are some other ways we can express this value?


0.𝟕𝟐


Example 3: Consider the following number: 0.𝟐𝟕 What are some other ways we can express this value?


0.𝟐𝟕

Writing Repeating Decimals in Rational FormWhiteboards

Talk Partners: Consider the following number: 3.𝟔 What are some other ways we can express this value?


Talk Partners: Consider the following number: 0.𝟏𝟖 What are some other ways we can express this value?


Talk Partners: Consider the following number: 0.2𝟔 What are some other ways we can express this value?

Writing Repeating Decimals in Rational FormFormative Journal Entry Colored Dots

Consider the following number: 2.𝟒 Express this number in rational form (as a fraction).

Show at least one other way to represent this value besides as a fraction.

Use your colored dots to show your confidence in your writing. Place the dot at the top of your paper.


Repeating Decimals (The TRICK)Journal Entry

If ALL DIGITS REPEAT: If all the digits repeat, the digits that repeat are in the numerator and for every digit that repeats, a 9 goes in the denominator. (Why: its just a cool number trickbecause math is COOL like that!)

Example 0.𝟖 = 𝟖

𝟗and 0.𝟐𝟕 =

𝟐𝟕

𝟗𝟗=

𝟑

𝟏𝟏(make sure to write your fraction in simplest

form)

If SOME OF THE DIGITS REPEAT: If some of the digit repeat, to find the numerator subtract the value of the digits that do not repeat from the value of all the digits after the decimal point. To find the denominator, place a 9 in the denominator for every digit that repeats then place a 0 in the denominator for every digit that does not repeat. Then write the fraction in simplest form.

Example: 0.41𝟔 416 (whole number) – 41(part that does not repeat) = 375 numerator6 – repeats (9) and 41 does not repeat (00) = 900 denominator

𝟑𝟕𝟓

𝟗𝟎𝟎=

𝟓

𝟏𝟐

Mathematical Practice Strategy:Friendly Talk Probes

Friendly Talk Probes is a strategy that involves discussion with your talk partner.

A scenario is given for you to read. Decide what you think about the statementindividually and write a brief summary.

Then you will turn and talk to your talking partner and decide if you agree

or disagree. If you agree, one of you will put a + on your whiteboard andhold it up. If you disagree, put a − on your whiteboard.


Probe 1: Two friends, Donald and Daisy, are discussing vocabulary words.

Donald says, “ I think some decimals are rational and some are irrational.

Daisy says, “ Donald, I think all decimals are rational because you can write alldecimal as fractions.”

Write down the name of the friend you most agree with. Explain why you agreewith that friend and not the other. If you do not agree with either friend, explain yourreasoning and show the correct answer.


Probe 2: Two friends, Mickey and Minnie, are discussing how to write the fraction𝟓

𝟏𝟏as a decimal.

Mickey says, “ To change a fraction to a decimal, we use long division. Here is thecorrect answer”:

Minnie, “ Hmm.. I think we may use long division to solve but Mickey I’m notsure your answer is correct, I think something might be wrong.

11.5

2.2

-10

1

0

0-10

0

Write down the name of the friend you most agree with. Explain your reasoning.If you agree with Minnie, decide what Mickey may have done wrong and find thecorrect answer.


Probe 3: Two friends, Chip and Dale, are discussing how to write the decimal1.𝟑𝟔 in rational form. Below are samples of their work.

Chip: 1. 𝟑𝟔 : Dale: 1. 𝟑𝟔 :

Let x = 0.363636…. Let x = 0.363636….

Multiply both sides by ten Multiply both sides by a hundred

10x = 3.66666.. 100x = 36.363636…

Subtract x from both sides Subtract x from both sides

10x – 1x = 3.6666… −.66666… 100x – 1x = 36.363636… −.363636…

9x = 3 99x = 36

Solve equation: 𝟗𝒙

𝟗= 𝟑

𝟗= 𝟏

𝟑so x =

𝟏

𝟑Solve equation:

𝟗𝟗𝒙

𝟗𝟗= 𝟑𝟔

𝟗𝟗=

𝟒

𝟏𝟏so x =

𝟒

𝟏𝟏

Solution 1𝟏

𝟑Solution = 1

𝟒

𝟏𝟏

Write down the name of the friend you most agree with. Explain your reasoning. If you don’t agree with either friend, show and explain what you think the correct answer is.

In your own words, describe the learning objective for learning target 1.

Learning Target 1 Class Debrief

What are some important vocabulary words we learned to use during target 1?

What are some different ways we can represent rational values?

Explain the process of writing the decimal expansion of a fraction.

Explain the process of writing a repeating decimal in rational form if1 digit repeats; more than one digit repeats; only if some of the digits repeat.

Explain the trick for writing repeating decimals in rational form.

What mathematical practices did we use in learning target 1?

What mathematical practices strategies/games did we use in target 1? Which strategies seemed to be the most helpful?


Learning Objective Target 2: I can solve and evaluate square roots of small perfect square and cube roots of small perfect cubes as solutions to

equations.

What does this symbol represent?

What operation does this symbol mean to do?

Number Sense Journal Entry

Geoboard ActivityAPP

http://illuminations.nctm.org/Activity.aspx?id=6385

Mathematical Practices Strategy:Silent Teach

In the Silent Teach Strategy, everyone is silent, includingthe teacher.

Using an array of colors, I will silently write some things on the board. Look for patterns with the numbers as well as the colors.I will then pass off a colored marker to you and I would like youto finish the pattern involving that color only.

If you are unsure of the answer, please put a question mark on the board, I will continue the pattern to see if you can find it. If not, Iwill allow you to pass the marker to someone else and will come back to you for another question.

Mathematical Practices Strategy: Silent Teach

Perfect Squares/Square Roots

What does this symbol represent?

What operation does this symbol mean to do?

Number Sense Journal Entry

Cube Towers Task

Activity 1: With the cubes, build the smallest cube tower. The length of the sides of the

tower should measure 1 unit. To find the volume of the tower, count the number of small cubes used to build the tower. How many cubes did you need to build this tower?

Activity 2: With the cubes, build the next smallest cube tower. The length of each of the sides of the tower

should measure 2 units. To find the volume, count the number of small cubes used to build the tower. How many cubes did you need to build the tower?

Activity 3: With the cubes, build the next smallest cube tower. The length of each of the sides of the tower

should measure 3 units. To find the volume, count the number of small cubes used to build the tower. How many cubes did you need to build the tower?

Make a prediction of the volume of the tower if the side length of the tower was 6, 4, and 8 units.Think and write about what mathematical practices were used during this task.

Cube Roots# of CubesCube Root # of CubesCube Root

Cube Root

Perfect Square/Cube Root EquationsGuided Notes

In this lesson we will be using our knowledge of perfect square and cube rootsto solve equations.

Remember an equation is a like a balance scale, everything must be equal on both sides.

In most equations, we are solving for a unknown value called a variable.

The variable is often represented with a letter such as x.

To solve an equation means to find every number that makes the equation true.

Mathematical Practices Strategy:Your in the Hot Seat

With the strategy, Your in the Hot Seat, a student name is drawn atrandom and comes up to the “Hot Seat” with their whiteboard and marker.

If you answer the question correctly, you get a prize and get to pull the nameof the next person to sit in the hot seat. Everyone else also answers on whiteboard.

If you are unsure of an answer, you may phone a friend to come up and whisper toyou the process of solve but you must be the person to give and explain the answer.After giving the answer, the phone a friend gets to choose the next person in thehot seat and you will each get a small prize.

If neither of you are sure of the answer, the teacher will draw a new person to be in the hot seat and no prize is awarded to first 2 participants.

Be thinking about what mathematical practices are being used during this activity.

Everyone else also answers on whiteboard. One random name will be drawn fromaudience. If you have it right, you also get a prize.

Example 1: Solve the given equation for x:

𝒙𝟐 = 121 for x


Example 2: Solve the given equation for y:

𝒚𝟐 = 𝟏𝟔

𝟏𝟔𝟗for y


Example 3: Solve the given equation for a:

𝒂𝟑 = 729 for a


Example 4: Solve the given equation for n:

𝒏𝟑 = 𝟖

𝟏𝟐𝟓for n


Example 5: Solve the given equation for z:

𝒛 = 5


Example 6: Solve the given equation for r:

𝒓 = 𝟐

𝟑


Example 7: Solve the given equation for p:

𝟑 𝒑 = 4


Example 8: Solve the given equation for g:

𝟑 𝒈 = 𝟔

𝟕



𝒙𝟐 + 4= 85



𝒙𝟑

𝟗= 3



2 𝒙 = 200



𝟑 𝒙

𝟐+ 2= 34


Formative Journal EntryColored Dots

Describe in your own words what a perfect square and what asquare root is. Give an example of this.

Describe in your own words what a perfect cube and what acube root is. Give an example of this.

Describe the numerical pattern we observed in the silent teach for perfectsquares/square roots.



Mathematical Practices Strategy:Think Aloud

In a Think A Loud, students work through a multi-step problem and talkthrough the process of finding a solution.

Each student first individually reads the problem and answers the following statements :

• The problem is asking….• The important information is ….• The strategy I will use to solve the problem is ….• The steps to using this strategy to solve the problem are ….• I know this answer makes sense because ….

After working individually, you will pair up with someone else and take turns goinggoing over the information you collected. Any ideas you both had or agree on, writein the agree column. Any ideas you have that you don’t quite agree on write in thedisagree column.

After working in pairs, you will then be assigned to a group and go through thethink allowed again. You will create a poster with your collected ideas to share.


Agree Disagree

Cubic ContainerThink Aloud

Problem Solving Task 1: A clear plastic container designed in the shape of a cube can have 216 square centimeter unit cube blocks placed inside of it to fill its volume. If I wanted to wrap this container with wrapping paper, what would be the surface area of the container? (Remember surface area is the total area of the outside of theobject)

• The problem is asking….

• The important information is ….

• The strategy I will use to solve the problem is ….

• The steps to using this strategy to solve the problem are ….

• I know this answer makes sense because ….


Agree Disagree

FarmingThink Aloud

Problem Solving 2: Farmer Bob is retiring from a 30 year career as a manager of a crowded, 500 acre farmwhere he raises livestock and poultry. His farm seems to be very crowded so he comes upwith a set of goals for ideal pasture size for his stock based on his years of experience and his beliefs about the grazing land required for specific species of animals. Here is a table describing his goals.

He decides to work on creating an ideal pasture space for his horses first based on his goals because his favorite animal is his horse named Lucky. Currently he owns 144 horses in total that he needs to make a space for.

If Farmer Bob decides to make his ideal space for his horses in the shape of a square and needs to put fencing around it to keep the horses enclosed, how much fencing will he need to purchase, in feet? (Hint one acre equals 43,560 feet)

chickens cattle goats horses sheep pigs

# per acre 50 12 20 4 10 25


Agree Disagree


Learning Target 2 Class Debrief


Explain the process of finding the square root of a number.

Explain the process of finding the cube root of a number

In the real world and other parts of math, where might these ideas be important?



UNIT 1: NUMBER SENSELearning Objective Target 3: I can identify rational and irrational numbers.

Rational and Irrational Numbers Video

https://www.youtube.com/watch?v=q_wstDWjnKQ

Classifying Numbers

Biologists classify animals based on shared characteristics.

For example, a cardinal is an animal, a vertebrate, a bird and a passerine (perching bird).

Classifying NumbersGuided Notes

We have learned that the set of rational numbers consists of whole numbers,integers and fractions.

The set of real numbers consists of rational and irrational numbers.

Irrational Numbers are numbers that are not rational. They can not be written in the form

𝒂

𝒃where a and b are integers and b is not 0. These types of numbers

produce decimals that do NOT terminate and do NOT repeat.

Square roots and cube roots of numbers that are not perfect squares or cubesare irrational. (Why do you think this may be the case?)

0.37485…

0.1234…

Formative Journal EntryRational and Irrational Numbers

Colored Dots

Classify the given numbers into the following categories: whole, integer, rational and irrational. Some numbers may go in more than 1 category.

𝟐 0 −𝟏𝟐

𝟒𝟐𝝅 0.567… 0.48736 0.121221222… 0.4545... 2. 𝟕 0.1𝟑



Example 1: Write all the names that apply to the given number:

𝟐



−𝟏𝟕. 𝟖𝟒



𝟎



𝟖𝟏

𝟑



𝟎. 𝟓𝟔𝟕𝟖𝟗…



− 𝟏𝟔𝟗



𝟑𝟏𝟎𝟎



A baseball pitcher has pitches 12𝟐

𝟑innings.


Example 8: Identify the set of numbers that best describes each situation. Justify your reasoning.

the number of people wearing glasses in a room.

the total area of the classroom, in meters.

the temperature on any given day in Alaska

the square root of any prime number

the total number of cheeseburgers you buy from McDonalds


In a Card Sort, you will sort cards based on given criteria.

Be listening to your partner/group justifications, you may be asked to share with the class their justifications as well as your own.

Mathematical Practices Strategy:Card Sort

When sorting with a partner or group, be able to justify to them your reasoning for wanting to sort cards in a certain way. Any cards you may disagree on, put them in a separate pile and use a post it note to label them disagree so they can be discussed as a class.

Mathematical Practices Strategy:Card Sort

Sort 1: Sort the cards into groups based on how your group feels they should be sorted. There can be as many groups as you like. Justify your reasoning behind your groups.

Sort 2: Sort the cards into2 groups: One group rational; one group irrational. Be able to justify why each card was placed into its given group.

Sort 3: Sort the cards into the following groups: whole numbers, integers, rational and irrational. Use the post it notes to make copies of numbers if you feel a number belongs in more than 1 group.

3 Way Tie/Concept Map: With the given words below, create connections of 3 ideas individually in your journal. Then you will be grouped in a team to share all your ideas to see if you can cross connect your ideas.

rational number irrational number decimal expansion repeating decimal

terminating decimal long division whole number integer fraction

percents square root cube root perfect square root perfect cube root

rational approximation

Mathematical Practices Strategy: 3 Way Tie / Vocabulary Concept Mapping

Reflectional Piece (Individual)Three Way Tie

Pick three types of categories of numbers we have studied and complete a threeway tie.

Concept Map Math Examples

Mathematical Practices Strategy: 3 Way Tie / Vocabulary Concept Mapping

Class Example

Reflective Journal Entry

What Mathematical Practices were used in this Strategy?

Was the 3 Way Tie/ Vocabulary Concept Map a beneficial strategy in learning vocabulary? Why/ Why not?

Why is learning vocabulary important in mathematics?


Learning Target 3Class Debrief


Define the categories of real numbers.

Explain how you can determine if a number is whole? An integer? rational?irrational?





Learning Objective Target 4: I can use rational approximation of irrational numbers to estimate, compare and locate irrational numbers on a number line to the nearest tenth.

Estimating Irrational Numbers Video

https://learnzillion.com/lessons/224-place-nonperfect-square-roots-between-2-integers

Rational ApproximationGuided Notes

We have learned that irrational numbers are numbers that can not bewritten as the ratio of 2 numbers. (cannot be written as fractions)

We can estimate where we may find them on a number line by usingrational approximation.

These types of numbers we have learned create decimals that do NOT terminate and do NOT repeat. Thus we can not find an EXACT locationfor them on a number line since we will never know how to round themcorrectly since we will not know what the next number in the decimal is.

Rational approximation is when we use rational numbers that we can findon a number line to approximate the location of irrational numbers near them.We usually do this to an accuracy of the nearest tenth or hundredth.

Our Perfect Square Roots

Rational ApproximationNumber Line Activity

Rational Approximation: To the Nearest TenthJournal Entry

Example 1: Estimate the value of 𝟐 to the nearest tenth. Then construct a number line and place a dot on its approximate location.

Rational Approximation: To the Nearest TenthWhiteboard

Talk Partner: Estimate the value of 𝟏𝟎 to the nearest tenth. Then construct a number line and place a dot on its approximate location.


Example 2: Estimate the value of − 𝟏𝟏𝟎 to the nearest tenth. Then construct a number line and place a dot on its approximate location.

Rational Approximation: To the Nearest TenthWhiteboards

Talk Partner: Estimate the value of − 𝟐𝟏 to the nearest tenth. Then construct a number line and place a dot on its approximate location.


Example 3: Estimate the value of 2 𝟓𝟎 to the nearest tenth. Then construct a number line and place a dot on its approximate location.

Rational Approximation: To the Nearest TenthWhiteboards

Talk Partners : Estimate the value of 4 𝟒𝟎 to the nearest tenth. Then construct a number line and place a dot on its approximate location.


Example 4: Estimate the value of 𝟐𝟔

𝟐to the nearest tenth. Then construct a number line

and place a dot on its approximate location.

Rational Approximation: To the Nearest TenthWhite Boards

Talk Partners: Estimate the value of 𝟔𝟐

𝟐to the nearest tenth. Then construct a number

line and place a dot on its approximate location.


Example 5: Estimate the value of 3𝝅 to the nearest tenth. Then construct a number line and place a dot on its approximate location.


Talk Partners: Estimate the value of 2𝝅 to the nearest tenth. Then construct a number line and place a dot on its approximate location.


Example 6: Estimate the value of 𝝅

𝟑to the nearest tenth. Then construct a number line and

place a dot on its approximate location.


Talk Partners: Estimate the value of 𝝅

𝟐to the nearest tenth. Then construct a number line

and place a dot on its approximate location.

Rational Approximation: Spoons Tournament

To play spoons, you will have two decks of colored cards, pink and orange and a setof 4 spoons.

First shuffle the deck of pink cards and pass out 3 cards to each person. These cards haveirrational numbers on them. You will have 10 minutes to find the rational approximationof these to the nearest tenth. Write your answers on a scratch sheet of paper or a whiteboardbut keep your answers hidden.

Now the dealer picks a card from the orange deck. If the card is equivalent to one of the cardsin his/her hand, he/she keeps it and it makes a pair that he/she keeps in her hand. If not, they discard it to the next person and this continues around the table. You may never have morethan 8 cards in your hand at anytime.

The first person to have all 3 matches grabs a spoon from middle of table. Everyone left must then also try to grab a spoon. The person left without a spoon is eliminated from the game.

If you feel the person’s matches are incorrect, you can challenge them by calling teacher overto check. If any pairs are incorrect, they automatically lose and other 3 members are winners.

Reflective Journal Entry…..

What were some of the benefits of playing the spoon game in helping reinforceLearning Target 4? What were some helpful strategies in helping you be successful in the game?

What were some of the challenges of playing spoons in helping reinforce LearningTarget 4? What were some things you really needed to be good at to be successful?

Did you think the spoons game was beneficial in helping you learn the material forTarget 4 Rational Approximation? Why/Why Not?


Learning Target 4Class Debrief


Describe the process of finding the rational approximation of an irrationalnumber involving a square root to the nearest tenth.

Describe the process of finding the rational approximation of an irrationalnumber involving Pi to the nearest tenth.




Unit 1: Number Sense ReviewMathematical Practices Strategy:

Friendly Corrective Feedback

You will be put into groups and each person will receive a paper with an assessment question and students work on it.

Work the problem out on your own piece of paper. If you agree with the student’s work/answer place a green dot on the top of your paper and staple it to the student work. Write one to two sentences justifying why you agree with their answer. If you disagree with their answer, place a red dot on the top of your paper.Write one to two sentences justifying why you disagree with their answer.

You will then pass the paper around. You will look at the next paper and repeat the process. You will then also look at your group’s dotted paper; answers and leave them feedback with a colored marker on their work.

Once you have worked through all the problems and you receive your original problem back, go through the problems again and read the feedback your groupleft on your answer. Then you will write a reflectional piece.

Mathematical Practices Strategy: Friendly Corrective Feedback

Reflective Journal Entry

What is your current level of confidence with each target?Place a colored sticker the resembles your confidence level next to each target listed. Pink (I do not understand at all) Orange (I understand some parts) Green(I understand most/all parts)

Describe your thoughts on the corrective feedback strategy. Was it helpful in reviewing for the common assessment? Why/Why Not

What is something you will go home and do this evening to help you prepare forthe common assessment tomorrow?

What is something I can do tomorrow to help you feel comfortable in taking theassessment?

What is something you can do different to be better prepared in the next unit?

What is something I can do better in helping you feel better prepared in the next unit?

Documents

Unit 1: THE Number system U1...Cube Towers Task Activity 1: With the cubes, build the smallest cube tower. The length of the sides of the tower should measure 1 unit. To find the volume