Grade 2 Demonstrate that changing the orientation of an object does not alter the measurements of its attributes [C R V]

Grade 5

8 Identify and describe a single transformation including a translation rotation and reflection of 2-D shapes [C T V]

9 Perform concretely a single transformation (translation rotation or reflection) of a 2-D shape and draw the image [C CN T V]

Grade 6

6 Perform a combination of translations rotations andor reflections on a single 2-D shape with and without technology and draw and describe the image[C CN PS T V]

7 Perform a combination of successive transformations of 2-D shapes to create a design and identify and describe the transformations[C CN T V]

8 Identify and plot points in the first quadrant of a Cartesian plane using whole number ordered pairs [C CN V]

9 Perform and describe single transformations of a 2-D shape in the first quadrant of a Cartesian plane (limited to whole number vertices)[C CN PS T V]

Grade 6 Angelo

Specific Outcomes4 Demonstrate an understanding of probability bybull identifying all possible outcomes of a probability experimentbull differentiating between experimental and theoretical probabilitybull determining the theoretical probability of outcomes in a probability experimentbull determining the experimental probability of outcomes in a probability experiment bull comparing experimental results with the theoretical probability for an experiment [C ME PS T]

I would not separate these outcomes There is no understanding of theoretical probability without doing experiemnts to see Coins and die When you flip a coin you have a choice of heads or tails A mathematician or scientist would sya that is a 50 50 chance There are 2 outcomes You get one or the other There is 1 of 2 results going to happen If you flip a coin a few times and keep track of the result you might get more heads than tails This is an experiemental result that is true for the number of times you flipped But it does not chance the facts that you had a 5050 chance of tails Flip hundreds of times save and compare the data You should see the optios even out to 5050 If they do not keep flipping

Any particular card is 1 of 52 in the deck If you turn a card over your chances of any specific card is 1 out of 52 This will take much longer to experiemtn with since there are so many possible outcomes

If you roll a die There are 6 possible outomes The possibility of any specific number coming up is 1 in 6 If you roll 6 times one of them is likely to be the 3 If it is not keep rolling Over time you should see the possibilities even out

So you might walk away saying heads come up more oten that tails However there are 2 possibilities on each flip and therefore a 50 50 chance of eight heads of tails being the result Because people do not have the time to flip hundreds of times they may fool themselves into thinking that heads is more likely but there is no logical reason why heads should be the result more often than tails When there are 2 choices and all other conditions are kept static there is a 50 50 chance of heads or tailsIf you repeat the flipping hundreds or even thousands of times you will see the data balance outThese outcomes link to the idea of fair testing in science When you remove all the variables but one there is no reason why you should not see 50 50 People tend to believe their immediate experiences but not gather enough data to really test out if what they think is happeing is really true

We also tend as humans to believe in ldquomagicrdquo or ldquoluckrdquo and while we may be willing to believe there is However if you continue to flip the coin over thousands and thousands of times you will begin to see

It All Depends (or Does It)To know why it is necessary to understand the difference between dependent and independent events For a dependent event its probability depends on the events leading up to it For example the probability of drawing the Ace of Spades from a deck of cards is 152 If you dont draw it the first time and discard whatever you drew instead there are now only 51 cards left so the probability goes up to 151 If you keep drawing cards and discarding them the probability of drawing the ace goes up until you get it Rolling dice is an independent event No matter how many times you roll and no matter what you get the probability of rolling a seven with two dice is 16 Whether you throw ten sevens in a row or dont get a seven for 20 straight rolls the probability is always 16

Recency BiasThe opposite of the gamblers fallacy is called recency bias Human beings have a unique tendency to subliminally put more weight on events that happened in the recent

past If events are independent it doesnt matter how many times in a row you flip heads with a fair coin The next flip is a 5050 shot to come up heads again

How to Identify Common Probability MisconceptionsYoure watching your friend Mickey play a dice game He rolls two dice and if he rolls a seven he wins The last five rolls hes gotten a two a four a five an eight and a twelve Its starting to get grim and from the way that big guy with the scar is looking at him youre starting to get a little worried for Mickey You tell him maybe its time to pack it in but hes somehow become more confident than before hes going to get a seven this time Dont you see He says my luck is bound to change soon Thats what the probability says You arent so sure hes right But then sure enough he rolls a seven I knew it he shouts Was he right all along

The Gamblers FallacyEven though Mickey won this time you were right to think he was crazy to be so sure his luck would change But first lets see why he was so sure his luck would change First Mickey realized there are six ways to roll a seven with two dice (16) (25) (34) (43) (52) and (61) Then he reasoned there are 6 sides x 6 sides = 36 possible combinations of numbers That means the probability of Mickey rolling a seven is 636 or 16 Mickey reasoned that if he had already rolled five times and not hit a seven the sixth time must be the charm Although it actually sounds pretty reasonable Mickey has committed the gamblers fallacy Even though he won his luck never actually changedTheoretical probability is a method to express the likelihood that something will occur It is calculated by dividing the number of favorable outcomes by the total possible outcomes

Theoretical vs Experimental Probability

When asked about the probability of a coin landing on heads you would probably answer that

the chance is frac12 or 50 Imagine that you toss that same coin 20 times How many times would you expect it to land on heads You might say 50 of the time or half of the 20 times So you would expect it to land on heads 10 times This is the theoretical probability

The theoretical probability is what you expect to happen but it isnt always what actually happens The table below shows the results after Sunil tossed the coin 20 times

The experimental probability of landing on heads is

It actually landed on heads more times than we expected

Now Sunil continues to toss the same coin for 50 total tosses The results are shown below

Now the experimental probability of landing on heads

is

The probability is still slightly higher than expected but as more trials were conducted the experimental probability became closer to the theoretical probability

Examples1 Use the table below to determine the probability of each number on a number cube

Lets Review Theoretical probability is what we expect to happen where experimental probability is what actually happens when we try it out The probability is still calculated the same way using the number of possible ways an outcome can occur divided by the total number of outcomes As more trials are conducted the experimental probability generally gets closer to the theoretical probability

httpwwwsoftschoolscommathtopicstheoretical_vs_experimental_probability

Grade 456

Demonstrate an understanding of ratio concretely pictorially and symbolically [C CN PS R V]

2 of these to one of these

4 of these to 2 of these

Use the rods

2 to one is frac12 4 to 2

3 to one is 13

It is 3 times as long 4 times as tall 6 times longer

Any particular card is 1 of 52 in the deck If you turn a card over your chances of any specific card is 1 out of 52 This will take much longer to experiemtn with since there are so many possible outcomes

If you roll a die There are 6 possible outomes The possibility of any specific number coming up is 1 in 6 If you roll 6 times one of them is likely to be the 3 If it is not keep rolling Over time you should see the possibilities even out

So you might walk away saying heads come up more oten that tails However there are 2 possibilities on each flip and therefore a 50 50 chance of eight heads of tails being the result Because people do not have the time to flip hundreds of times they may fool themselves into thinking that heads is more likely but there is no logical reason why heads should be the result more often than tails When there are 2 choices and all other conditions are kept static there is a 50 50 chance of heads or tailsIf you repeat the flipping hundreds or even thousands of times you will see the data balance outThese outcomes link to the idea of fair testing in science When you remove all the variables but one there is no reason why you should not see 50 50 People tend to believe their immediate experiences but not gather enough data to really test out if what they think is happeing is really true

We also tend as humans to believe in ldquomagicrdquo or ldquoluckrdquo and while we may be willing to believe there is However if you continue to flip the coin over thousands and thousands of times you will begin to see

It All Depends (or Does It)To know why it is necessary to understand the difference between dependent and independent events For a dependent event its probability depends on the events leading up to it For example the probability of drawing the Ace of Spades from a deck of cards is 152 If you dont draw it the first time and discard whatever you drew instead there are now only 51 cards left so the probability goes up to 151 If you keep drawing cards and discarding them the probability of drawing the ace goes up until you get it Rolling dice is an independent event No matter how many times you roll and no matter what you get the probability of rolling a seven with two dice is 16 Whether you throw ten sevens in a row or dont get a seven for 20 straight rolls the probability is always 16

Recency BiasThe opposite of the gamblers fallacy is called recency bias Human beings have a unique tendency to subliminally put more weight on events that happened in the recent

past If events are independent it doesnt matter how many times in a row you flip heads with a fair coin The next flip is a 5050 shot to come up heads again

How to Identify Common Probability MisconceptionsYoure watching your friend Mickey play a dice game He rolls two dice and if he rolls a seven he wins The last five rolls hes gotten a two a four a five an eight and a twelve Its starting to get grim and from the way that big guy with the scar is looking at him youre starting to get a little worried for Mickey You tell him maybe its time to pack it in but hes somehow become more confident than before hes going to get a seven this time Dont you see He says my luck is bound to change soon Thats what the probability says You arent so sure hes right But then sure enough he rolls a seven I knew it he shouts Was he right all along

The Gamblers FallacyEven though Mickey won this time you were right to think he was crazy to be so sure his luck would change But first lets see why he was so sure his luck would change First Mickey realized there are six ways to roll a seven with two dice (16) (25) (34) (43) (52) and (61) Then he reasoned there are 6 sides x 6 sides = 36 possible combinations of numbers That means the probability of Mickey rolling a seven is 636 or 16 Mickey reasoned that if he had already rolled five times and not hit a seven the sixth time must be the charm Although it actually sounds pretty reasonable Mickey has committed the gamblers fallacy Even though he won his luck never actually changedTheoretical probability is a method to express the likelihood that something will occur It is calculated by dividing the number of favorable outcomes by the total possible outcomes

Theoretical vs Experimental Probability

When asked about the probability of a coin landing on heads you would probably answer that

the chance is frac12 or 50 Imagine that you toss that same coin 20 times How many times would you expect it to land on heads You might say 50 of the time or half of the 20 times So you would expect it to land on heads 10 times This is the theoretical probability

The theoretical probability is what you expect to happen but it isnt always what actually happens The table below shows the results after Sunil tossed the coin 20 times

The experimental probability of landing on heads is

It actually landed on heads more times than we expected

Now Sunil continues to toss the same coin for 50 total tosses The results are shown below

Now the experimental probability of landing on heads

is

The probability is still slightly higher than expected but as more trials were conducted the experimental probability became closer to the theoretical probability

Examples1 Use the table below to determine the probability of each number on a number cube

Lets Review Theoretical probability is what we expect to happen where experimental probability is what actually happens when we try it out The probability is still calculated the same way using the number of possible ways an outcome can occur divided by the total number of outcomes As more trials are conducted the experimental probability generally gets closer to the theoretical probability

httpwwwsoftschoolscommathtopicstheoretical_vs_experimental_probability

Grade 456

Demonstrate an understanding of ratio concretely pictorially and symbolically [C CN PS R V]

2 of these to one of these

4 of these to 2 of these

Use the rods

2 to one is frac12 4 to 2

3 to one is 13

It is 3 times as long 4 times as tall 6 times longer

past If events are independent it doesnt matter how many times in a row you flip heads with a fair coin The next flip is a 5050 shot to come up heads again

How to Identify Common Probability MisconceptionsYoure watching your friend Mickey play a dice game He rolls two dice and if he rolls a seven he wins The last five rolls hes gotten a two a four a five an eight and a twelve Its starting to get grim and from the way that big guy with the scar is looking at him youre starting to get a little worried for Mickey You tell him maybe its time to pack it in but hes somehow become more confident than before hes going to get a seven this time Dont you see He says my luck is bound to change soon Thats what the probability says You arent so sure hes right But then sure enough he rolls a seven I knew it he shouts Was he right all along

The Gamblers FallacyEven though Mickey won this time you were right to think he was crazy to be so sure his luck would change But first lets see why he was so sure his luck would change First Mickey realized there are six ways to roll a seven with two dice (16) (25) (34) (43) (52) and (61) Then he reasoned there are 6 sides x 6 sides = 36 possible combinations of numbers That means the probability of Mickey rolling a seven is 636 or 16 Mickey reasoned that if he had already rolled five times and not hit a seven the sixth time must be the charm Although it actually sounds pretty reasonable Mickey has committed the gamblers fallacy Even though he won his luck never actually changedTheoretical probability is a method to express the likelihood that something will occur It is calculated by dividing the number of favorable outcomes by the total possible outcomes

Theoretical vs Experimental Probability

When asked about the probability of a coin landing on heads you would probably answer that

the chance is frac12 or 50 Imagine that you toss that same coin 20 times How many times would you expect it to land on heads You might say 50 of the time or half of the 20 times So you would expect it to land on heads 10 times This is the theoretical probability

The theoretical probability is what you expect to happen but it isnt always what actually happens The table below shows the results after Sunil tossed the coin 20 times

The experimental probability of landing on heads is

It actually landed on heads more times than we expected

Now Sunil continues to toss the same coin for 50 total tosses The results are shown below

Now the experimental probability of landing on heads

is

The probability is still slightly higher than expected but as more trials were conducted the experimental probability became closer to the theoretical probability

Examples1 Use the table below to determine the probability of each number on a number cube

Lets Review Theoretical probability is what we expect to happen where experimental probability is what actually happens when we try it out The probability is still calculated the same way using the number of possible ways an outcome can occur divided by the total number of outcomes As more trials are conducted the experimental probability generally gets closer to the theoretical probability

httpwwwsoftschoolscommathtopicstheoretical_vs_experimental_probability

Grade 456

Demonstrate an understanding of ratio concretely pictorially and symbolically [C CN PS R V]

2 of these to one of these

4 of these to 2 of these

Use the rods

2 to one is frac12 4 to 2

3 to one is 13

It is 3 times as long 4 times as tall 6 times longer

The experimental probability of landing on heads is

It actually landed on heads more times than we expected

Now Sunil continues to toss the same coin for 50 total tosses The results are shown below

Now the experimental probability of landing on heads

is

The probability is still slightly higher than expected but as more trials were conducted the experimental probability became closer to the theoretical probability

Examples1 Use the table below to determine the probability of each number on a number cube

Lets Review Theoretical probability is what we expect to happen where experimental probability is what actually happens when we try it out The probability is still calculated the same way using the number of possible ways an outcome can occur divided by the total number of outcomes As more trials are conducted the experimental probability generally gets closer to the theoretical probability

httpwwwsoftschoolscommathtopicstheoretical_vs_experimental_probability

Grade 456

Demonstrate an understanding of ratio concretely pictorially and symbolically [C CN PS R V]

2 of these to one of these

4 of these to 2 of these

Use the rods

2 to one is frac12 4 to 2

3 to one is 13

It is 3 times as long 4 times as tall 6 times longer

Lets Review Theoretical probability is what we expect to happen where experimental probability is what actually happens when we try it out The probability is still calculated the same way using the number of possible ways an outcome can occur divided by the total number of outcomes As more trials are conducted the experimental probability generally gets closer to the theoretical probability

httpwwwsoftschoolscommathtopicstheoretical_vs_experimental_probability

Grade 456

Demonstrate an understanding of ratio concretely pictorially and symbolically [C CN PS R V]

2 of these to one of these

4 of these to 2 of these

Use the rods

2 to one is frac12 4 to 2

3 to one is 13

It is 3 times as long 4 times as tall 6 times longer

Every minute I can throw 5 How many can I throw in 3 minutes

Ratio table

Grade 5 Equivalent fractions

httpswwwillustrativemathematicsorgcontent-standards6RPAtasks2157

ldquoAndrew and Sam are running equally fast around a track Andrew started first When he had run 9 laps Sam had run 3 When Sam had run 15 how many had Andrew completedrdquo

13 becomes 2sixths For every thrd there are 2 sixths So 5 thirds would be 10 sixths why

Proportional Reasoning involves understanding that Equivalent ratios can be created by iterating andor partitioning into a composed unit If one quantity in a ratio is multiplied or divided by a factor then the other quantity must be multipled or divided by the same factor to maintain the proportional relationship

A company makes charms for bracelets For every 3 hearts it makes 2 diamonds If the company makes 15 hearts how many diamonds does it make

Cuisenaire rods to find the rods32 ratio bull Build a train of 3 red rods and build a one-color train using only 2 rods underneath the red train bull Next build a train of 9 red rods and build the green train with the same length bull Finally build a train of 15 red rods and build the corresponding green train

I buy 3 books for $12 How much will 15 books cost

The ratio of benches to trees in a park is 24 If there are 12 trees how many benches are there in the park

Each spring Paul and his family go to Grandpas farm to pick strawberries Paul eats 2 strawberries for every 9 strawberries he puts in his basket If Paul ate 8 strawberries how many strawberries did he put in his basket

A herring swims 3 kilometers in 30 minutes Another day the herring swims 7 kilometers in 70 minutes Is this a proportional relationshipUse centimeter grid paper to determine if the relationship is proportional If 20 people are ahead of you in the lunch line it takes 12 minutes to get your lunch If 30 people are ahead of you it takes 18 minutes Is the relationship proportional

David and Sara Basic facts for division

The outcomes that mention division in Grade 3

Grade 3 relating multiplication to division

Demonstrate an understanding of division (limited to division related to multiplication facts up to 5 1113089 5) by

representing and explaining division using equal sharing and equal grouping

creating and solving problems in context that involve equal sharing and equal grouping

modelling equal sharing and equal grouping using concrete and visual representations and recording the process symbolically

relating division to multiplication [C CN PS R]

Understand and recall division facts related to multiplication facts to 5 x 5

Grade 4Describe and apply mental mathematics strategies to determine basic multiplication facts to 9 x 9 and related division facts [C CN ME R]

Understand and apply strategies for multiplication and related division facts to 9 x 9 Recall multiplication and related division facts to 7 x 7

This is what it means to understand division 20divide 4 means I have 20 books and I want to share them onto 4 shelves How many books per shelf Or 20 divide 4 means I have 20 books and I want to put 4 on each shelf how many shelves will I put books on Without this understanding it does not really matter if students remember the facts as they do not understand them

The outcomes about division all say related to multiplication Students must understand there are 2 ideas in division equal groups and equal shares They need to see and understand both in arrays

Understanding comes before recallhellip because recall means you understand and can retrieve from memory or solve by strategizing

The outcomes is clear they must understand and recall Division is the inverse it must be related to multiplication in order to make sense

Communication is our goal we need to focus on how students explain how division is related to multiplication the two ways to interpret a division situation and create and explain contexts that make both interpretations evident

factors product divisors quotients equal groups fair sharing

Students will read a division equation then place it in a context 24 divide 6 = I have 24 books Six can fit on each shelf How many shelves will I use I know 6 x 4 = 24 so I will need 4 shelves

I have 24 books and I want to share them equally among 6 kids How many books will each kid get

The Big Idea concept based curriculum

When learners are able to think multiplicatively they can apply the commutative property the associative property the distributive property and inverse relations to solve problems

Kouba (1989) Steffe (1992) Lorway (2018) Mulligan and Mitchelmore (1997) and Mulligan amp Watson (1998)

6 12 18 24 vs I give everyone 2 thatrsquos 12 books gone so another 2 each

• It All Depends (or Does It)
• Recency Bias
• How to Identify Common Probability Misconceptions
• The Gamblers Fallacy
• Theoretical vs Experimental Probability

A herring swims 3 kilometers in 30 minutes Another day the herring swims 7 kilometers in 70 minutes Is this a proportional relationshipUse centimeter grid paper to determine if the relationship is proportional If 20 people are ahead of you in the lunch line it takes 12 minutes to get your lunch If 30 people are ahead of you it takes 18 minutes Is the relationship proportional

David and Sara Basic facts for division

The outcomes that mention division in Grade 3

Grade 3 relating multiplication to division

Demonstrate an understanding of division (limited to division related to multiplication facts up to 5 1113089 5) by

representing and explaining division using equal sharing and equal grouping

creating and solving problems in context that involve equal sharing and equal grouping

modelling equal sharing and equal grouping using concrete and visual representations and recording the process symbolically

relating division to multiplication [C CN PS R]

Understand and recall division facts related to multiplication facts to 5 x 5

Grade 4Describe and apply mental mathematics strategies to determine basic multiplication facts to 9 x 9 and related division facts [C CN ME R]

Understand and apply strategies for multiplication and related division facts to 9 x 9 Recall multiplication and related division facts to 7 x 7

This is what it means to understand division 20divide 4 means I have 20 books and I want to share them onto 4 shelves How many books per shelf Or 20 divide 4 means I have 20 books and I want to put 4 on each shelf how many shelves will I put books on Without this understanding it does not really matter if students remember the facts as they do not understand them

The outcomes about division all say related to multiplication Students must understand there are 2 ideas in division equal groups and equal shares They need to see and understand both in arrays

Understanding comes before recallhellip because recall means you understand and can retrieve from memory or solve by strategizing

The outcomes is clear they must understand and recall Division is the inverse it must be related to multiplication in order to make sense

Communication is our goal we need to focus on how students explain how division is related to multiplication the two ways to interpret a division situation and create and explain contexts that make both interpretations evident

factors product divisors quotients equal groups fair sharing

Students will read a division equation then place it in a context 24 divide 6 = I have 24 books Six can fit on each shelf How many shelves will I use I know 6 x 4 = 24 so I will need 4 shelves

I have 24 books and I want to share them equally among 6 kids How many books will each kid get

The Big Idea concept based curriculum

When learners are able to think multiplicatively they can apply the commutative property the associative property the distributive property and inverse relations to solve problems

Kouba (1989) Steffe (1992) Lorway (2018) Mulligan and Mitchelmore (1997) and Mulligan amp Watson (1998)

6 12 18 24 vs I give everyone 2 thatrsquos 12 books gone so another 2 each

• It All Depends (or Does It)
• Recency Bias
• How to Identify Common Probability Misconceptions
• The Gamblers Fallacy
• Theoretical vs Experimental Probability

Understanding comes before recallhellip because recall means you understand and can retrieve from memory or solve by strategizing

The outcomes is clear they must understand and recall Division is the inverse it must be related to multiplication in order to make sense

Communication is our goal we need to focus on how students explain how division is related to multiplication the two ways to interpret a division situation and create and explain contexts that make both interpretations evident

factors product divisors quotients equal groups fair sharing

Students will read a division equation then place it in a context 24 divide 6 = I have 24 books Six can fit on each shelf How many shelves will I use I know 6 x 4 = 24 so I will need 4 shelves

I have 24 books and I want to share them equally among 6 kids How many books will each kid get

The Big Idea concept based curriculum

When learners are able to think multiplicatively they can apply the commutative property the associative property the distributive property and inverse relations to solve problems

Kouba (1989) Steffe (1992) Lorway (2018) Mulligan and Mitchelmore (1997) and Mulligan amp Watson (1998)

6 12 18 24 vs I give everyone 2 thatrsquos 12 books gone so another 2 each

• It All Depends (or Does It)
• Recency Bias
• How to Identify Common Probability Misconceptions
• The Gamblers Fallacy
• Theoretical vs Experimental Probability