Teacher Name: Gauri Pathak Grade & Section:11 C Subject: Mathematics
Unit Name: Statistics(Representation of Data (S1) ,Measures of Location (S1),Permutation and combination)
Duration for completion of Unit: 20 days
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About the Unit: In this unit students will cover topics based on statistics which comes under AS Level Mathematics S1 exam which includes (Representation of Data (S1) ,Measures of Location (S1) Measure of spread(S1) Probability). In Representation of Data. Students will learn difference between qualitative and quantitative data and will learn how to display numerical data using diagrams. In Measures of Location students will learn three different measure of locating mean ,mode, median. In Probability student will understand about calculating probability using different ways.
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Where the Unit fits in: This unit will help students to Develop their mathematical skills in learning Statistics in a way which encourages confidence in solving CIE problems. It will help to acquire a range of Statistical skills , particularly those which will enable them to use in applications of statistics in the context of everyday situations and of other subjects they may be studying .It will also help to acquire the mathematical background necessary for further study in this or related subjects. It Covers ~30 % of total S1 exam paper. It aims to teach students the basics of statistics , so that when you move onto the later units (Statistics) you will have a basic understanding.
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Prior Learning: It is helpful if the students have the knowledge of Statistics Basics before .So one class will be taken as to revise Statistical Fundamentals and discussing future learning .
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Future Learning:
One class will be taken to make them understand how they can use Statistics in real life .
Students can use their knowledge and understanding to understand
Mean
• It helps teachers to see the average marks of the students.
• It is used in factories, for the authorities to recognize whether the benefits of the workers is continued or not.
• It is also used to contrast the salaries of the workers.
• To calculate the average speed of anything.
• It is also used by the government to find the income or expenses of any person.
• Using this the family could balance their expenses with their average income.
Applications of Mode
• Mode contains the highest frequency in any data. Its applications areas follows:-
• It is used to influx of the public transport.
• The no. of games succeeded by any team of players.
• The frequency of the need of infants.
• Used to find the number of the mode is also seen in calculation of the wages, in the patients going to the hospitals, the mode of travel etc
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Expectations:
At the end of this unit:(Based on CIE Requirements)
Most students will be able to
comprehend translations of common realistic contexts into mathematics; use the results of calculations to make predictions, or comment on the context; and, where appropriate, read critically and comprehend longer mathematical arguments or examples of applications;
.Students will be able to use use mathematical models and analyze change in both real and abstract contexts.
AO2 construct rigorous mathematical arguments and proofs through use of precise statements, logical deduction and inference and by the manipulation of mathematical expressions, including the construction of extended arguments for handling substantial problems presented in unstructured form;
AO3 recall, select and use their knowledge of standard mathematical models to represent situations in the real world; recognise and understand given representations involving standard models; present and interpret results from such models in terms of the original situation, including discussion of the assumptions made and refinement of such models;
Some students will not have made so much progress will be able to recall, select and use their knowledge of statistical facts, concepts and techniques in a variety of contexts;
Some students will progress further will be able to derive understanding of the connections between different elements of the subject. They will be able to integrate and draw together of knowledge, understanding and skills learned in the unit in different parts of the A level course through using and applying methods developed at earlier stages of study in solving problems.
Based on Unit :
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Language for Learning:
Through the activities in this unit students will be able to understand, use and spell correctly academic/ new vocabulary words:
Student Personalized dictionary: Guide them to make a Personalized dictionary where everytime they learn a new vocabulary work in any exercise or unit they can keep a record in a diary or at the end of notebook.
Reading-students will learn close reading of the text-comprehend what is read and then Carry out a transformation from the words of the problem to the
selection of an appropriate mathematical strategy.
Writing –Students will learn how to Encode the answer in an acceptable written form.
Listening and Speaking- students will learn to prepare for and participate effectively in a range of conversations and collaborative activities in class.
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Resources: (Materials both print and electronic that we draw upon for our instruction)
Resources include collected past year problems based on this unit.
Post its ,Marker, ICT Lab,If possible Field trip to collect data,SUBWAY Worksheet ,Rubrics.
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Teaching& Learning Activities:
Day 2:
Do Now/Activator (15min): Start teaching by using the strategy called “Class Brainstorm” in which the students will be asked to give as many ideas as
possible about the term “Data”.Show them a chart which display
Which will make them understand vocabulary related to Data Collection.
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Mini Lesson (15 min):Then the students will be explained terminology related to representation of data and how to draw stem and leaf diagram for combining the raw data with the help of examples.
Differentiation: Class is divided into groups such that each group has a leader who explains the stem and leaf diagram to his peers (peer explanation).
Learning Activity (25 min):
The students will draw stem and leaf diagram related to the problems on page no. 8 of book “Statistics I” by Steve Dobbs and Jane Miller .
Teacher Role: Observation of individual student involvement in class with a walk around while students are drawing the diagrams.
Wrap Up (10 min): Have students draw a mind map of the steps they learned for stem leaf diagram.
Out of school learning:
Students will complete their Mind Maps given based on stem and leaf and will also solve problems given from book.
Day 3:
Do Now/Activator (10min): HW Problems will be discussed .Students will bring their mind maps and will explain about it.
Mini Lesson (15 min): Reminding students of what they have learnt in previous lesson connecting to this lesson by using strategy “Teaching by asking”. The students will be asked some of the questions related to histograms and cumulative graphs.
Differentiation: Teacher help will be given by a short description of the topics to late bloomers.
Learning Activity: Collect and visualize data as Histograms:
· Have students collect data that can be continuous such as height. Record their height in inches on a White Board.
· Draw and label an X- and Y-axis on the board with categories of classification that are as wide as a White Board.
· Ask your students to place their Post-it Note in the correct category, stacking them as they go.
· Immediately everyone will see the shape of the histogram.
· Sample:
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Learners will make a histogram and cumulative frequency graphs for a given data.
1. Then Students will solve problems based on CIE past papers.
2. “Thumbs up and down” strategy will be used to assess whether the students understand the question or not
Wrap Up (10 min): Give each pair of students a picture of a stop light. In the green box, they write something they already knew about today’s topic. In yellow, they write something they learned today and in red, they write something they didn’t understand about what was discussed and discuss what was not understood.
Out of school learning:
Students generate word problems: Have students create word problems based on histogram. Through the construction of a problem the students learn what to look for when solving word problems they are assigned. Students will solve CIE problems based on topic .
Day 4:
Do Now/Activator (10min): Discuss about problems which were given as homework and if any queries will be rectified in the beginning of the class.
Mini Lesson (15 min ) Recall all the terms and methods for Representation of data .
Using explicit explanations of the steps of problem solving through teacher demonstrating the thought process used in problem solving taking few problems based on Past questions in CIE .
Learning Activity (25 min): The students will solve questions 2,4 (pg16) & questions 2,4 (pg 18) of book “Statistics I” by Steve Dobbs and Jane Miller
Stratergy:Reciprocal peer tutoring (RPT) to improve math achievement: having students pair, choose a team goal to work toward, tutor each other on math problems, and then individually work a sheet of drill problems. Students get points for correct problems and work toward a goal.
Teacher’s Role: Teacher observation glancing at the work done by the students.
Wrap Up (10 min): Key Ideas :Students list the key ideas from the lesson and ways to solve it.
Out of school learning: Students will create Graphing Anchor Chart as a Group .
Sample:
Day 5:
Do Now/Activator (10min): Collect their charts and display and ask them to explain about it.
Two Minute Talks
To activate prior knowledge and focus ,student learning on the topic will be addressed. Students will share with a partner by brainstorming everything they already know (prior knowledge) about the topic. In doing so, they are establishing a foundation of knowledge in preparation for learning new information about the topic.
Mini Lesson (15 min): Introduce the three different measures of location (mean, median and mode) taking few examples on board.
Discuss few problems based on Past questions in CIE .
Differentiation: Giving late bloomers the formula required- mean, median or mode.
Learning Activity (25 min):
1. A discussion on each “Measures of location” and relating them to a given situation will follow, with each student sharing an example of each method.
2. The students will take notes while discussion.
Formative Assessment through student and teacher interaction while discussion.
Wrap Up (10 min): Have students line up at the door at the end of the period and as they leave they must share an idea or concept they learned. Each student must give a different answer.
Out of school learning:
Students will create A Web based Project of their own choice based on Application of (Mean ,Mode ,Median) in real life. Before selecting a project they are suggested to do research on Application of Mean Mode Median in real life. 3 days will be given to complete the project (Permission will be taken form Ritu to use Computer Lab or students can work on their Laptops by taking permission ).They can go to nurse or Account office after school (After taking permission) to work on project depending on their choice on project .
Example :
They can search on work on How mean median mode is used in Medical Fields.
They can search on work on How mean median mode is used in Economics.
They can look for how mean median mode can be used in Miri Piri Academy or any Institute ……….etc. (can ask for help if required)
I want them to come up with idea how can they use Mean ,Median ,Mode which ever field they select .It will suggested them to come up with real data collection . Using any one application they create a ppt ,chart ,excel sheet or anything they find suitable(they will work in groups assigned by teacher)
Day 6:
Do Now/Activator (10min): “Quick write” Tell students to write whatever comes to mind about the previous topic, without regard to written conventions—a brainstorming on paper for a limited amount of time.
Mini Lesson (15 min):
· Teacher will be Connecting the three different measures of location (mean, median and mode) through a quick class quiz.
· Then some examples based on median will be solved on the board.
· Discuss few problems based on Past questions in CIE .
Differentiation: Giving the formula of median as required according to the odd or even frequency.
Learning Activity (25 min): The students will take notes of the examples and then solve questions of pg. no. 26
Wrap Up (10 min): Exit Ticket : Write formula for median on exit cards
Out of school learning:
:Students will continue working on their project .
Day 7:
Do Now/Activator (10min): Post “An Average Rhyme” on the door before they enter class just to create interest In the topic.
Entrance Ticket : Students will be asked questions based on Mean ,Mode which will help to bring background knowledge to the forefront, make connections with the previous day’s lesson
Mini Lesson (15 min):
1. Relation between the median and mode will be explained.
2. Formulas will be given .
3. Discuss few problems based on Past questions in CIE .
Learning Activity (25 min): Students will take notes Everyone will be given some different problems to solve . Students will work at different levels individually in one classroom.
CIE past papers problems will be given after deconstructing.
Peer assessment followed by the teacher assessment.
Wrap Up (10 min):Students will discuss their answers and problem they faced while solving problems.Students will leave giving a verbal answer about what they learnt today.
Out of school learning:Students will continue working on their project.
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Day 8:
Do Now/Activator (30 min): Assigned project will be collected and dicussed.
Mini Lesson (10 min): Recall all the measures of location by comparing and contrasting the three measures of location for ungrouped data.
A model will be given which has the comparison of mean , median and mode and then students are asked to solve the miscellaneous question which is based on all three measures of location.
Differentiation:The learners will be explained by breaking the miscellaneous question into parts using strategy “CUBES” .
Learning Activity (10 min): The students will learn the comparison of three measures of location and solve question no. 2 of pg.no. 36.Discuss few problems based on Past questions in CIE based on stratergy
Signal cards : Students use a card to indicate their level of understanding of a concept. Cards may be labeled as follows:
Yes/ No
Wrap Up (10 min): At the end of the lesson, students should write What’s still confusing me …..Ask students to provide a brief, written identification of what topics they consider to be confusing and not fully understood.
Out of school learning:
Day8: Expert to Expert:Students will be given some problems where Students will work collaborative to teach each other keywords and decoded terms and formulas determined by the teachers.
Day 9:
Do Now/Activator (10min): Work on their doubts .Their collaborative work will be discussed and then Use a brief video clip(on Laptop) on a pertinent topic Range and quartile to activate student interest in the focus of an upcoming lesson.
Mini Lesson (15 min): Introducing the different measures of spread (range and quartiles) and sharing examples with the students to find the interquartile range from the given quartiles.
the diagrams
Learner Activity(25 min): The students will be divided into groups such that each group has a leader who explains the range and interquartile range to his peers (peer explanation) .
The students will take notes of the examples and then solve q.no.6,7 (pg.48)
Teacher’s role: Observation of individual student involvement in class with a walk around while students are drawing
Wrap Up (10 min): Discuss the key points of the topic.
Out of school learning:
: They will solve 3 Past paper problems will be given after helping them in Deconstructing.
Day 10:
Do Now/Activator (10min): Address their queries in their homework
Mini Lesson (15 min):
1. Reminding students of what they have learnt in previous lesson connecting to this lesson by using strategy “Teaching by asking”. The students will be asked the steps of finding the inter quartile range.
2. Box and Whisker plot will be explained to the students.
3. Discuss few problems based on Past questions in CIE based on Box and Whisker plot.
Learning Activity (25 min ) : The students will understand the Box and Whisker plot by taking notes and then solve q.no.10 (pg.49)
“Thumbs up and down” strategy will be used to assess whether the students understand the question or not.
Wrap Up (10 min): Give each pair of students an index card. Ask them to write out everything they can remember about the day’s content they like .
Out of school learning:
Students will be given some problems based on Box and Whisker plot.
Day 11:
Do Now/Activator (10min): Discuss about problems which were given as homework and if any queries, will be rectified in the beginning of the class. Plus/Minus/Intriguing :The teacher should write prompts, controversial statements, etc. (related to the focus of the upcoming lesson) ahead of time to which students will be asked to respond. Have them identify the statements they agree with (+), the statements they disagree with or question (-), and the statements they find intriguing (I).
Mini Lesson (15 min): Review of previous day’s work by asking students the comparison of histograms and box and whisker plots and is followed by the discussion.
Differentiation: Teacher help will be given by a short description of the topics.
Learning Activity (25 min). The students will give reasons whether box and whisker plots or histograms are better for comparing two distributions. Students will solve Problems 2problems from past papers.
Formative assessment based on the discussion.
Wrap Up (10 min):Verbal Exit Ticket based on their learning Experience in class for that day .
Out of school learning:
Read for the next topic “Variance and Standard Deviation”.
Day 12:
Do Now/Activator (10min): Give students the key word which will be addressed in the lesson. Ask them to write a detail or descriptor with which they are already familiar that starts with each of the key word.
Mini Lesson (15 min):
Connecting to previous day’s homework Examples of finding Standard deviation will be given.
Giving them the formula required- standard deviation.
Learning Activity (25 min):
· The students will take notes of the examples and then solve q.no.1-3 (pg.53)
· Peer assessment
Wrap Up (10 min):Taken as learning Activity.
Out of school learning:
Book Miscellaneous Questions will be given After deconstructing by themselves.
Day 13:
Do Now/Activator (10min): Discuss about problems which were given as homework and if any queries will be rectified in the beginning of the class. Explain a Procedure :Ask students to Explain how to Derive standard deviation in data giving an example based on previous day work.
Mini Lesson (15 min): Demonstrating the way to calculate the variance from a frequency table and sharing two different formulas .
Formative Assessment: Marking students’ work
Learning Activity (25 min): .The students will apply the given formulas to do Q.no. 1,3,5,7,8 (pg.57) Rubrics will be given with Problem solving instruction: explicit instruction in the steps to solving a mathematical problem including understanding the question, identifying relevant and irrelevant information, choosing a plan to solve the problem, solving it, and checking answers.
Wrap Up (10 min): Problems approaches will be discussed based on rubrics.
Out of school learning:
Summary of what are the steps to solve various ways of variance in given data and problems 1CIE problem to solve after decoding in class.
Day 14:
Do Now/Activator (10min): Collect summary Discuss about problems which were given as homework and if any queries will be rectified in the beginning of the class.
Pose a question based on their previous knowledge that What might be the definition of probability?
Mini Lesson (15 min): Introducing the concept of probability by explaining the related terminology (assigning probability, probabilities of events and the addition law for mutually exclusive events). Sharing some of the examples of all the above mentioned concepts
Discuss few problems based past CIE papers.
Learning Activity (25 min): The students will take notes of the examples and then solve q.no.1-5 (pg.71)
Carousel Brainstorming: On chart paper around the room (or on paper that is passed around groups), ask small groups of students (3-4) to respond to a question or statement based on few real life examples posed at the top of the paper.
Wrap Up (10 min): Discuss answers of assigned problems and how they have approaches to particular way of solving it.
Out of school learning:
: Students will be assigned Past papers problems based on Probablility.
Day 15:
Do Now/Activator (10min): Discuss about problems which were given as homework and if any queries will be rectified in the beginning of the class. Pose a question or specific aspect of focus for the upcoming topic which is Conditional probability. Ask students to “quick write” (write whatever comes to mind about the topic, without regard to written conventions—a brainstorming on paper) for a limited amount of time. Begin with short time periods of 1-2 minutes, because students must write the entire time.
Students will be explained how to interpret something
Mini Lesson: Describing the students the laws about the conditional probability and independent events(using Tree Diagrams ).
The problems will be explained after discussing it.
“Thumbs up and down” strategy will be used to assess whether the students understand the question or not.
Learning Activity (25 min): The students will take notes of the examples and then solve q.no.1,2,3 (pg.79)
Wrap Up (10 min): Key Ideas
Students list the key ideas from the lesson and why they were important.
Out of school learning:
Problems will be given from text book based on taught topic.
Day 16:
Do Now/Activator (10min): Discuss about problems which were given as homework and queries will be rectified in the beginning of the class. Ask to create questions based on yesterday’s learning. In response to a prompt or question, direct students to “turn and talk” to a shoulder partner
Mini Lesson (15 min): Recall all the laws of conditional probability and independent events. The students will be asked questions related to previous day’s description.
Discuss few problems based on Past questions in CIE..
Learning Activity (25 min):
· The students will raise their hands if they know the answer and give answer .
· The answers given by one student will be discussed among the other students and after discussion will solve q.no.4-6 and 8-13 (pg.79).
· Marking students’ work.
· The problems will be explained by teacher after demystifying it.
Wrap Up (10 min): Sticky Notes Give students sticky notes and a question or topic to which they should respond. Have them post their notes on the board, door, wall, or a chart (that can have divisions.). Students will return to those responses at the end of the lesson to confirm or revise their thinking.
Out of school learning:
:CIE problems will be Given with a rubric based on their knowledge .
Day 17:
Do Now/Activator (10min): Discuss about problems which were given as homework and if any queries will be rectified in the beginning of the class. Word Splash Students are given a “splash” of the key words from the unit. They should write a few meaningful sentences to capture (relying on their background knowledge) using these words.
Mini Lesson (15 min):
Introduction to the “Permutations and combinations” by using Teaching by asking. The students will be asked about the ‘Factorial’ and how to calculate it .Then the teacher will relate the factorial with permutations taking examples of everyday life when the objects are distinct and describing the formula of permutation. Also explaining the permutation when the objects are not distinct.
Learning Activity (25 min): The students will answer the questions the teacher ask and learn about the permutations (application of factorials) and then take notes.
Students will be given some problems based on Past CIE after deconstructing.
Wrap Up (10 min): Exit Slip :Students will write main idea of concept they have
Out of school learning:
CIE past papers problems will be Given after Deconstructing few questions in class only with rubric (Assignment Attached) .
Day 18:
Do Now/Activator (10min): Discuss about problems which were given as homework and if any queries will be rectified in the beginning of the class.
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Mini Lesson (15 min):
Connection to previous day’s work. The questions given in previous day’s homework will be discussed. The teacher will continue by explaining the combinations through various examples.
Analyzing a Real World Problem through Permutations/Combinations:
Many restaurants out there are doing math and showing it off? I want my students to do the math and analyze the validity of these advertisements.
Divide number of students in group :1 group will work on Wendy burger
Wendy burger "We figured out that there are 256 ways to personalize a Wendy's hamburger. Luckily someone was paying attention in math class." Is it really so???? This is an old package which displays ways to personalize a Wendy's hamburger.
Discuss the number of toppings available at Wendy's (9 total) and then decide to use combinations to figure out the number of arrangements possible since order does not matter.and then check if its really true ????
Other group will work on Sonic
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Curriculum
S1
1. Representation of data:
· select a suitable way of presenting raw statistical data, and discuss
· advantages and/or disadvantages that particular representations may have
· understand and use different measures of central tendency (mean, median, mode) and variation (range, interquartile range, standard deviation), e.g. in comparing and contrasting sets of data
· use a cumulative frequency graph to estimate the median value, the quartiles and the interquartile range of a set of data
· understand and use different measures of central tendency (mean, median, mode) and variation (range, interquartile range, standard deviation), e.g. in comparing and contrasting sets of data
· use a cumulative frequency graph to estimate the median value, the quartiles and the interquartile range of a set of data
2. Permutations and combinations
· understand the terms permutation and combination, and solve simple problems involving selections
· solve problems about arrangements of objects in a line, including those involving:
· repetition (e.g. the number of ways of arranging the letters of the word ‘NEEDLESS’)
· restriction (e.g. the number of ways several people can stand in a line if 2 particular people must — or must not — stand next to each other).
3. Probability
· evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g. for the total score when two fair dice are thrown), or by calculation using permutations or combinations
· use addition and multiplication of probabilities, as appropriate, in simple cases
Learning Objectives: (Based on Cambridge)
Students
• understand relevant mathematical concepts, terminology and notation of Statisitcs ;
• recognise the appropriate mathematical procedure for a given situation;
• apply combinations of mathematical skills and techniques in solving problems;
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Assessment:
Formative Assessment: KWL
Formative Assessment :Explaining the Procedure (Learner Activity)
Formative Assessment: Problem solving using Rubric
Summative Assessment: Monthly test
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Learning Outcomes:
Reading-students will learn close reading of the text of given problem-comprehend what is read and then Carry out a transformation from the words of the problem to the selection of an appropriate mathematical strategy.
Writing –Students will learn how to Encode the answer in an acceptable written form. Students will be able to label Statistical graphs with appropriate units and scaling .
Listening and Speaking- Students will be able to communicate to others the results of statistical studies.
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1. How many different ways do you think there are to personalize a Sub ?
2. List out Different toppings and sauces to put on sub at SUBWAY??
3. TO calculate ,we will use permutation or combination .which one shoud we use ?why?
4. What is formula for combination?
5. What is formula for permutation?
6. Calculate the number of ways to make a sub based on topping that you come up with?
The inclusion of differentiated activities ensures that students of all levels can participate and benefit from the lessons. For students seeking additional support, math tuition can be a great way to reinforce these concepts and clarify any doubts. With personalized guidance, students can further develop their problem-solving skills and gain confidence in tackling complex problems. This blog post is an excellent resource for both teachers and students to enhance their understanding of these vital mathematical concepts.
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