Late level 4 plan (term 1)

Planning notes
This plan is a starting point for planning a mathematics and statistics teaching programme for a term. The resources listed cover about 50% of your teaching time. Further resources need to be added to meet the specific learning needs of your class, to support your local curriculum and to provide sufficient teaching for a full term.
Resource logo
Level Four
Integrated
Units of Work
This unit requires students to find relationships between variables and calculate probabilities. It is set in the context of exploring ideas related to whakataukī (proverbs).
  • Find simple exponential number patterns.
  • Use exponential patterns to make predictions about further members of the pattern.
  • Use tables and graphs to identify relationships between two variables.
  • Create rules for relationships between two variables.
  • Use fractions to measure probabilities.
  • ...
Resource logo
Level Four
Number and Algebra
Units of Work
This unit presents a range of strategies for solving multiplication and division problems with multi-digit whole numbers. Students are encouraged to notice the structure of problems, and to anticipate which strategies might be best suited to solving them. This unit builds on the ideas presented in...
  • Mentally solve whole number multiplication and division problems using:
    1. proportional adjustment
    2. place value partitioning
    3. rounding and compensation
    4. factorisation.
  • Use appropriate recording techniques.
  • Predict the usefulness of strategies for given problems.
  • Evaluate the...
Resource logo
Level Four
Geometry and Measurement
Units of Work
This unit explores important ideas about reflective and rotational symmetry.
  • Recognise when one figure is a reflection of another using variant properties such as perpendicular distance from the mirror line, equalities of lengths, areas and angles, and opposite orientation in relation to the mirror line.
  • Recognise the rotational symmetry of a figure including...
Resource logo
Level Four
Number and Algebra
Units of Work
This unit teaches students to identify linear relationships and solve linear equations in context.
  • Identify and find values for variables in context.
  • Identify linear relationships in context.
  • Represent linear relationships using tables, graphs and simple linear equations.
  • Draw strip diagrams to represent linear equations.
  • Solve simple linear equations and interpret the answers in...
Resource logo
Level Three
Number and Algebra
Units of Work
The purpose of this series of lessons is to develop understanding of equivalent fractions and the operations of addition and subtraction with fractions.
  • Add and subtract fractions with like denominators.
  • Explore and record equivalent fractions for simple fractions in everyday use.
  • Recognise that equivalent fractions occupy the same place on the number line.
  • Recognise and apply the multiplicative relationship between simple equivalent...
Source URL: https://nzmaths.co.nz/user/75803/planning-space/late-level-4-plan-term-1

Whakataukī

Purpose

This unit requires students to find relationships between variables and calculate probabilities. It is set in the context of exploring ideas related to whakataukī (proverbs).

Achievement Objectives
GM4-1: Use appropriate scales, devices, and metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time.
NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
NA4-9: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.
S4-3: Investigate situations that involve elements of chance by comparing experimental distributions with expectations from models of the possible outcomes, acknowledging variation and independence.
S4-4: Use simple fractions and percentages to describe probabilities.
Specific Learning Outcomes
  • Find simple exponential number patterns.
  • Use exponential patterns to make predictions about further members of the pattern.
  • Use tables and graphs to identify relationships between two variables.
  • Create rules for relationships between two variables.
  • Use fractions to measure probabilities.
Description of Mathematics

This unit spans number, algebra, measurement and statistics. Several important mathematical concepts are encountered:

  1. Relationships between variables.

Variables are measures of some feature or attribute. A variable might come from simply counting, such as finding the number of strands of spaghetti. Most variables come from measurements, such as the mass of an object or the time in seconds. Relationships are rules (functions) that connect two or more variables. For example, the number of handshakes possible in a group of five people is 4 + 3 + 2 + 1 = 2 x 5. Finding more data from different numbers of people might lead to a function rule for finding the number of handshakes, such as h = n(n-1)/2, where h is the number of handshakes and n is the number of people.

  1. Types of relationships

There is a multitude of different relationship types, but a few are privileged in primary and lower secondary school for two reasons; they are relatively simple and common in the real world.

  1. Linear relationships are called so because graphs of the relationship form straight lines.
  2. Quadratic relationships occur in situations where the value of one variable is related to the square of the other. Many situations in physics, such as the stopping distance of a car as a function of speed, are quadratic. The graphs of quadratics also model the paths of projectiles.
  3. Exponential relationships occur in situations where the value of one variable is constantly changing by a fixed ratio. Situations like population growth and bush fires can be modelled with exponential functions.
  1. Probability

Probability is a measure of the chance of an event occurring. Probabilities are a fraction comprised of the number of favoured outcomes out of the number of possible outcomes. For example, the chance of getting an even number with one dice roll is one half since there are three favoured outcomes (2, 4, 6) out of six possible outcomes (1, 2, 3 ,4, 5, 6) and 3/6=1/2.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:

  • Act out many of the situations using students in the class before using more abstract representations, such as diagrams, tables, and equations. Doing favours (Session One) and relationships (Session Four) are ideal for acting out. Other sessions such as Descendants (Session Two) and Kourā (Session Five) can be modelled with physical objects.
  • Use digital technology to supports students investigate the concepts in these sessions. Calculators should be used to empower students who find the calculation demands difficult. Graphing applications such as spreadsheets, and online tools, should be used, wherever possible, to afford greater attention to the concepts in play, and less attention to the technical demands of graphing. This is not to diminish the need for students to learn calculation and graphing skills at other times.
  • Explicitly model the performance of important skills as they are needed. Students need exposure to models of how to use measurement tools, to choose correct units, to create tables, and use patterns to seek generalisations.

The sessions in this unit draw on a limited set of whakataukī. Several online sites provide a large range of proverbs based around themes. Whatataukī typically employ metaphors to represent a moral purpose. A useful way to adapt the contents of this unit is to encourage students to choose a whakataukī that has significance to them. They can design an experience for their peers that captures the moral purpose of the whakataukī. Investigating the proverb, and it’s meaning, may serve many other areas of the curriculum, particularly literature, social and pure sciences, and wellbeing.

Required Resource Materials
  • Copymasters One, Two, and Three
  • PowerPoints One, Two, Three, and Four
  • Spreadsheet video
  • Dry strands of spaghetti, paper cups, pieces of string, duct tape, scissors, paper clips, weights (e.g. coins, sinkers, washers), scales (as sensitive as possible).
Activity

Session One

In this session we consider the mathematics of relationships among people, whanaungatanga, using whakataukī. (proverbs). Those relationships are also important for all creatures and physical objects in the real world.

He aroha whakatō, he aroha puta mai.
If kindness is sown, then kindness you shall receive

  1. Show the students PowerPoint One. Each day Tīpene must give a kindness to twice as many people as he did the day before.
    How hard will it be for Tīpene to complete the wero (challenge)?
  2. Let the students investigate these questions:
    How many people will Tīpene give a kindness to on each day of the week? Remember that he starts on Monday and sees Kuia again on Sunday.
    How many people will Tīpene do a kindness to in total from Monday to Sunday?
  3. Let students investigate the problem. You might allow calculator use for some students, but most calculations can be done mentally. Look for students to organise their data. A table is a good way to do that:
    Day

    Monday

    Tuesday Wednesday Thursday Friday Saturday Sunday
    Kindnesses 1 2 4 8 16 32 64
    Total 1 3 7 15 31 63 127
  4. Ask: What patterns do you see in the table that make predicting the numbers for the next day easy?
    Obviously, the number of kindnesses doubles from the previous day. You might discuss students’ strategies for doubling, e.g. adding to itself or multiplying by two.
    Do they notice that the total is one less than the number of kindnesses on the next day? For example, a total of 63 on Saturday is one less than the number of new kindnesses, 64, on Sunday.
  5. Ask: What will happen if Tīpene keeps going in this way?
    You might ask students to write the last few slides for Kuia and Tīpene (PowerPoint One). Is Tīpene able to complete his mahi? Does he get to play video games?
  6. Ask students to complete a graph of the number of new kindnesses related to the day number, using Copymaster One. The graph is exponential as there is a constant ratio of two between consecutive terms, e.g. 32 is twice 16 and 32÷16=2.
  7. Ask questions that relate to specific points on the graph, such as:
    On what day did Tīpene give 16 kindnesses?
    How many kindnesses did Tīpene give on Saturday (Day 6)?
  8. Discuss other phenomenon in the real world that grow exponentially. The number of people in Aotearoa has grown exponentially since 1858, allowing for the impact of wars and influenza. Graphs are easily accessed online. Spread of diseases like measles, growth of funds in a bank account, and uptake of mobile phone technology are other good examples. The ratios in these examples are seldom as tidy as two.
    There are about 5 million people in Aotearoa/New Zealand.
    How many days would Tīpene need to do kindnesses for, so everyone received a gift from him?
    Of course, he could never physically do that many kindnesses!
  9. Allow access to calculators and computers. A spreadsheet is an effective tool for continuing the above table until a total number of 5 000 000 people is reached. The problem can be solved algebraically but that is beyond students at this level. It would take Tīpene between 22 and 23 days to reach a total over 5 million kindnesses.
  10. Depending on the interest of your students you might explore questions like:
    • What would happen if Kuia asked Tīpene to treble the number of kindnesses each day?
    • How many days would it take him then to give a kindness to every kiwi?
    • How could Tīpene use other people to pay kindnesses to each other so he does no end up with an impossible task? (Search for the trailer to Pay It Forward online)
    • What is the major problem with the population of the world growing exponentially?

Session Two

The next investigation is an extension for capable mathematicians. Another whakataukī that can be used to illustrate exponential growth is:

Mēnā ka tiakina te kiwi he maha ngā uri
If protected, the kiwi has many descendants

  1. Use PowerPoint Two to introduce the problem. We have about 68 000 kiwi left but lose 2% of the unprotected birds each year, mainly due to other introduced animals like pigs, dogs, cats, stoats and rats. Female kiwi generally only lay only one egg in a clutch (litter) though the brown kiwi can lay 2-3 eggs. Being slow breeders puts the kiwi in danger of extinction. Preserving our native animals is an important aspect of kaitiakianga (guardianship). Care for the environment is also an important component of te ao Māori as it is for cultures throughout the world.
    Let’s assume there are only 50 fertile female kiwi left in New Zealand and they are all protected. There are also 50 fertile male kiwi as well. Half of the chicks that hatch are male and half are female.  Let’s say a female begins laying in her third year of life and she lays two eggs per year for 15 years. Find a way to predict the number of kiwi alive after 15 years, if we protect them all.
  2. Start the students off by considering the first three years. Draw a table like this and discuss as you fill it in:
    What will the table look like at the end of Year 1? Look for students to interpret the conditions and suggest the appropriate numbers to go in the table.
    Discuss why the table doesn't have columns for males (half the chicks are female and half are male, so you can just double the female population to find the total population).
    Year

    Breeding females

    Females born this year

    Young females

    Total females

    Total population

    1 50 50 50 100 200
               

     

  3. Ask students to create their own table and extend it.
    What will the table look like at the end of Year 2? 
    Year

    Breeding females

    Females born this year

    Young females

    Total females

    Total population

    1 50 50 50 100 200
    2 50 50 100 150 300
               

     

  4. Ask them to extend their table for the third year.
    Why will the table get more complicated in Year three? The 50 young females from Year 1 will start to breed.
    Students should apply the condition that new females lay eggs in their third year of life.
    Year

    Breeding females

    Females born this year

    Young females

    Total females

    Total population

    1 50 50 50 50 200
    2 50 50 100 150 300
    3 100

    100

    150 250 500
               
  5. If 100 more females were born this year, why are there only 150 young females? (The first 50 chicks are now old enough to breed.)

  6. Ask your students to create a spreadsheet to model the growth in the kiwi population for 15 years. Encourage them to use formulas and then fill down the columns. Click to download an example spreadsheet.
    • Breeding females = the number of breeding females from the previous year plus the number of females born two years previously
    • Females born this year = the number of breeding females
    • Young females = the females born this year and the previous one
    • Total females = the number of breeding females plus the number of young females
    • Total population = double the number of females
  7. Discuss what the spreadsheet shows.
    How many kiwi would there be after 15 years?
    The spreadsheet shows that if totally protected, 100 kiwi would become over 150 000 in 15 years. It is an impressive result but shows that conservation efforts can have dramatic effects.
  8. Discuss any false, or unrealistic assumptions that have been made for this exercise:
    • It assumes that no kiwi die
    • It assumes that all of the 50 breeding females at the start continue to breed for the full 15 years
    • It assumes that all females of breeding age breed successfully every year.
  9. Students might also be interested in the famous survival story of the Chatham Island black robin.
    https://www.nzonscreen.com/title/seven-black-robins-1981
    https://youtu.be/MUgpl5X6_xo
  10. Students might also be interested in the effect of changes to the conditions. For example:
    What happens to the population prediction if 10% of kiwi dies of natural cause each year?
    What happens to the population prediction if the fertility rate changes from two eggs per year to three?

Session Three

Whiria te tangata
Weave the people together

  1. Discuss the whakataukī. What do your students think it means? It is usually interpreted to mean “Unity is strength.”
    Do you know of situations where unity works?
  2. Students might suggest situations like friends sticking together to achieve a task or confront a threat, or when many people worked together to change views on an issue. Cyclists and runners work together in a pack to make the task easier (less drag). Animals also work together, e.g. birds fly in a formation. A search for “meerkats strength in numbers” will return videos where the meerkats collectively ‘see off’ a cobra.
    A related whakataukī is:
    Mā tini mā mano ka rapa te whai.
    By many, by thousands, the work will be accomplished (Many hands make light work)
  3. Show students a single strand of spaghetti.
    Imagine this strand of spaghetti represents the strength of a single person. A single strand may not be strong, but many strands working together may hold a lot of weight.
  4. Use Slides One to Nine of PowerPoint Three to introduce the experiment. Discuss what is meant by tensile strength (the ability of an object to withstand stretching force). Each group of students will need:
    Dry strands of spaghetti, paper cup, piece of string, duct tape, scissors, paper clip, weights (e.g. coins, sinkers, washers), scales (as sensitive as possible).
  5. Let students carry out the investigation in pairs or threes. Encourage them to record their data systematically and use it to make predictions.
    • Do students test the tensile strength consistently or are there variations to the conduct of the experiment? (Concept of fair testing)
    • Do students measure the mass required to break the spaghetti correctly and interpret the measures? (e.g. 183 grams)
    • Do students create tables of data and use pattern to make interpolations and extrapolations from the known data?
  6. After a suitable period bring the class together to share results. You might use Slides ten and eleven of PowerPoint Three to show some data presented as a scatterplot.
    What is represented on each axis? (x axis shows strands of spaghetti (independent or explanatory variable) and the y axis shows mass upon breaking (dependent or response variable))
    Is there a linear relationship between the two variables? (The first graph suggests linear but the second does not)
  7. Discuss the line of best fit (regression line) as going through the centre of the data. Students might enter their own data into a spreadsheet, or online graphing tool, and get the software to calculate the line of best fit.
  8. Return to the whakataukī.
    Whiria te tangata
    Weave the people together
    What does the experiment suggest about the importance of unity?

Session Four

Another whakataukī about relationships is:

Ehara taku toa i te toa takitahi, engari he toa takitini
My strength is not from myself alone, but from the strength of the group

This whakataukī is about the fact that part of the strength of a group is the number of interpersonal connections that can be made. The relationship between the size of the group and the number of connections is not linear.

  1. Go outside to a concrete area where you can draw with a piece of chalk. Ask three students to stand as the corners of an equilateral triangle. Ask another student to draw all the possible two person relationships among the three students. Three relationships are possible.

    How many two person relationships are possible with five people?
  2. Let students talk in pairs to create a prediction. Some students may think that five relationships will be possible since three were possible with three people. Form a regular pentagon with five students and get another student to draw all the two person connections.

    What shape is formed by the connection lines? (a pentagram inside a pentagon)
    What is the total number of two person connections? (10)
    How many connections does each person make? (Four)
    If 5 x 4 = 20, why are there only ten connections? (Each connection can be in two directions, A to B and B to A)
  3. Go inside and provide the students with Copymaster Two. The investigation aims at finding a general rule for the number of two person connections with n people. After a suitable time gather the class to share their ideas.
    The completed table should look like this:
    Number of people 1 2 3 4 5 6 7 8
    Number of connections 0 1 3 6 10 15 21 28

     

  4. Discuss patterns in the table. Ideas might include:
    • The differences between terms in the bottom row increase by one each time, i.e. 0 + 1 = 1, 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, …
    • Each number of connections is the sum of connections for the previous number of people and the previous number of people, e.g. 10 = 6 + 4, 28 = 21 + 7. This rule can be written recursively as Tn=T(n-1)+(n-1).
    • Each number of connections is half of the number of people multiplied by the number of people less one, e.g. 15 equals 1/2 × (6 × 5), and 21 equals 1/2 × (7 × 6).
  5. The last pattern can be expressed as a general rule: “Take the number of people, multiply it by one less than the number of people, then halve the result.”
  6. Algebraically this rule can be written as c=p(p-1)/2, where p represents the number of people and c represents the number of connections.
    Why does the rule involve multiplying the number of people by one less than the number of people? (Each person can connect with each of the other people in the group)
    Why is the number of connections halved? (Each connection is counted twice)
  7. To extend the ideas further you could:
    • Work on the unit called The truth about triangles and squares.

    • Investigate a different pattern in the connections within a group of people between ‘distant’ friends. Suppose five people stand as corners of a pentagon. Each person connects only to those people not immediately next to them. The connections would be:

      This is like finding all the diagonals of a polygon.
      Imagine six people. The distant friend connections are:

      Each person has p - 3 = 4 connections to distant friends. That makes 7 x 4 = 28 one-way connections. Half of 28 equals 14, the number of two-way connections.
      In general, if p represents the number of people, then the number of connections, c, is given by c=p(p+3)/2.

Session Five

In this session students confront their bias towards subjective judgment in situations involving chance. A good example of subjectivity is that students often believe that six is harder to get on a standard dice roll, than the other numbers one to five. Researchers believe that this bias is linked to games where it is crucial to roll a six, e.g. to escape jail in Monopoly. The urgency of getting a particular outcome affects people’s perceptions of how easy it is to get that outcome.

  1. Begin with this traditional whakataukī:
    He manako te kōura i kore ai.
    There are no crayfish as you set your heart on them.
  2. Ask: What do you think this whakataukī means?
  3. Let students share their interpretations. The proverb means that trying too hard to get something you want can be counter-productive
    I have a normal dice here. My Kuia (Grandmother) says if I do all my mahi I can roll this dice. If it lands on ono (six) I get crayfish for tea. Is that likely to happen?
  4. Discuss the likelihood of getting ono (six). Take a poll. Let students choose which number they think will come up. Discuss the poll results. For example:
Tau Tahi
(1)
Rua
(2)
Toru
(3)
Whā
(4)
Rima
(5)
Ono
(6)
Frequency 5 3 6 4 7 2

 

  1. Look for patterns in the table. Is ono the least popular choice? Why? Why not?
    Example:
    Why do so many of you think tahi, toru or rima are most likely, and I won’t get crayfish?
  2. Roll the dice once to see what happens.
    Is it just luck, or can I say what my chance of getting crayfish is?
    In the case of a single dice six outcomes are possible; 1, 2, 3, 4, 5 or 6. My desired outcome, ono, has a ‘one out of six’ chance of occurring. You might go to one of the many dice simulators online and look at what occurs with a sample of 120 dice rolls. If students have digital devices each member of the class might carry out 120 virtual rolls.
  3. Are the distributions (shape of the data) the same for everyone? Why? Why not?
  4. Show the students how to play ‘Kōura Deal or No Deal.’ This is a gameshow from overseas simplified with a kiwi flavour. The object of the game is to get as many kōura (crayfish) as possible to feed your whānau.
    Will you find a way to get what you desire? – lots of crayfish!
    Or will the whakataukī prove correct?
    Using Copymaster Three students cut out many paper kōura. Students also need opaque plastic cups, labels A, B, C, D, E for the tops of the cups, and cards labelled 0, 1, 2, 4, 8, also made from Copymaster Three. You might use Slide One of PowerPoint Four to illustrate the game. Without the PowerPoint on show you can edit the slides which includes moving them around.
    There are two people in the game, the guest and the host. A turn goes like this:
    1. Host offers the guest a number of kōura to quit the game, e.g. “I offer two kōura.”
    2. The guest either accepts the offer or plays on by choosing a cup, e.g. “I choose C.”
    3. The cup is turned over to reveal how many kōura (counters) are under it. That rock and the matching number card are removed. The game finishes either when the guest accepts an offer and gets the offered number of kōura (counters), or there is one cup remaining. In that case, the number of kōura under the remaining cup is the prize.
  5. Let students play the game. Ask them to swap roles between guest and host consecutively. Remind them that when they are guest their objective is to get as many kōura as possible. As host their role is to minimalise the number of kōura they give away. Students should keep track of how many kōura they win.
  6. Look for these features:
    • Do students use subjective judgment to make their decisions? ,e.g. “I am always unlucky so I will take the offer.”
    • Do they consider the probability of losing or winning when making their decisions, when either choosing or offering.
    • Do they make offers that are close to the average of the numbers of kōura left?
  7. After a suitable time gather the students to discuss the strategies they used to maximise the number of kōura. In a sense, there are no right or wrong decisions since this is a game of chance. Expect students to note that their decision depends a lot on what happens previously. This is called conditional probability, when a previous event impacts on the probability of a future event.
    What are your chances of picking four cups in a row that leave only the eight kōura? (one fifth)
    What are your chances of getting at least two kōura? (three of the five cups cover two or more kōura, so the chances are three fifths.
  8. Use Slides Two to Four of PowerPoint Four to discuss what decisions should be made in different scenarios. In each case ask:
    Would you take the offer, or play on? Why?
  9. With each slide you can simulate the game continuing, assuming the offer is declined. Exiting show on the PowerPoint allows you to move cups to reveal how many kōura are underneath. Scallop number cards can also be removed.
  10. With each scenario discuss:
    What are you chances of getting more than what is offered, if you play on?
    • On Slide Two there is a one third chance of getting four kōura, but a two thirds chance of getting two or more kōura.
    • On Slide Two there is a one half chance of getting more than the offered number of kōura. Is it worth the risk?
    • On Slide Three there is a one quarter chance of getting more than the offered number of kōura. However, there is a three quarters chance of eliminating one of the low numbered rocks in your next turn. Do you take the risk?
      ​​​​​​​Is the offer reasonable? Explain why.
  11. Try to determine a way to decide if the offer is reasonable. Should the offer be the average of the numbers left? Should it be less than the average? (In the actual gameshow offers are much less than the average in the early rounds.)
  12. Return to the original whakataukī. Do you think this whakataukī is true? Does wanting something badly make it less likely to happen?
Attachments

Multiplication and Division Pick n Mix 1

Purpose

This unit presents a range of strategies for solving multiplication and division problems with multi-digit whole numbers. Students are encouraged to notice the structure of problems, and to anticipate which strategies might be best suited to solving them. This unit builds on the ideas presented in the Multiplication Smorgasbord session in Book 6: Teaching Multiplication and Division.

Achievement Objectives
NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
NA4-8: Generalise properties of multiplication and division with whole numbers.
Specific Learning Outcomes
  • Mentally solve whole number multiplication and division problems using:
    1. proportional adjustment
    2. place value partitioning
    3. rounding and compensation
    4. factorisation.
  • Use appropriate recording techniques.
  • Predict the usefulness of strategies for given problems.
  • Evaluate the effectiveness of selected strategies.
  • Generalise the types of problems that are connected with particular strategies.
Description of Mathematics

The New Zealand Curriculum requires students to understand and use a range of mental, written and digital calculation strategies to multiply and divide multi-digit whole numbers. This unit of work is useful for students working at or towards Level Four Stage 7-Advanced Multiplicative of the Number Framework). Students at this stage partition and recombine numbers to simplify calculations and draw on their knowledge of multiplication facts and related division facts with factors up to ten. Understanding whole number place value underpins all strategies in this unit.

The key teaching points are:

  • Features of problems, particularly the numbers involved, privilege the efficiency of particular strategies. Teachers should elicit strategy discussion about problems in order to encourage students to justify their decisions about strategy selection in terms of the usefulness and efficiency of the strategy for the given problem situation.
  • Useful strategies for multiplication include place value partitioning, rounding and compensating, proportional adjustment and factorisation.
  • Useful strategies for division include proportional adjustment (with factorisation), rounding and compensating, and partitioning or ‘chunking’.
  • Tidy number strategies (rounding and compensating) are useful when number(s) in an equation are close to an easier number to work from. For example, 6 x 48 might be solved using 6 x 50 = 300.
  • When applying tidy numbers in multiplication and division it is important to keep track of what has been changed in a problem in order to compensate (rounding and compensating). For example, 6 x 2 must be subtracted from 300 to get the product of 6 x 48.
  • Standard place value partitioning is always a trustworthy strategy that is particularly appropriate where one or both factors are not easily rounded up. For example, 6 x 43 is best solved as 6 x 40 + 6 x 3.
  • Proportional adjustment is useful when there is a ‘common factor’ connection between the factors in multiplication, or dividend and divisor in division, that can be used to simplify the problem such as doubling and halving or quadrupling and quartering. Division with factorisation can be viewed as a form of proportional reasoning. In division both the dividend and divisor must be adjusted by the same factor.
Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

  • Using place value-based materials alongside symbols to develop both understanding and fluency with calculation methods.
  • Developing supportive algorithms that allow students to take progressive steps towards an answer, e.g., The ladder method of division.
  • Encouraging students to work in collaborative teams to develop explanation and justification strategies.
  • Varying the size and complexity of numbers in the problems to cater for a range of proficiencies.

The contexts for this unit include cycling, working at a fruit shop, transporting people to netball, rowing crews, and delivering pamphlets. Other situations of relevance to your students might be used to capitalise on contextual knowledge and motivate ākonga. For example, fundraising for an event, preparing a class feast, and teams in waka ama may provide useful story shells.

Required Resource Materials
Activity

Getting Started

The purpose of this session is to explore the range of strategies that your students already use to solve multiplication and division problems. This will enable you to evaluate which strategies need to be focused on in greater depth as well as identifying students in your group as "expert" in particular strategies.

Problem 1 (Copymaster Two):
Vanessa bikes 38 kilometres each day for five days. How many kilometres has she travelled by the end of the five days?

Ask students to work out the answer in their head if they can and record their strategy on paper. Give the students an appropriate amount of thinking time. Then ask them to share their solutions with their learning partner. The following are possible responses:

Note that the convention is to record the multiplier first so equations should be written as 5 x 38 =  .

Rounding and compensating:
5 x 38
38 is rounded to 40 so the problem becomes 5 x 40 = 200, then 10 (5 x 2) is subtracted from the product to get 5 x 38 = 190.

In full the strategy might be written as 5 x 38 = 5 x 40 – 5 x 2.

View how to do this using Video One.

Proportional adjustment: 
5 x 38
Solve instead 10 x 19 using doubling and halving (by doubling 5 and halving 38).

In full the strategy might be written as 5 x 38 = 10 x 19

View how to do this using Video Two.

Place value partitioning: 
5 x 38
Solve 30 x 5, add 8 x 5.

The strategy can be written as 5 x 38 = 5 x 30 + 5 x 8 or in working form:

View how to do this using Video Three.

As different strategies arise ask the students to explain why they chose to solve the problem in that way. Accept all the strategies that are elicited at this stage, recording them to reflect upon later in the unit (perhaps in a modelling book). Be aware that some students may elect to add rather than multiply.

For example:

                      

You might like to discuss the efficiency of multiplication versus repeated addition.

Problem 2 (Copymaster Three):
There are 256 rowers entered in the eights rowing champs at the Maadi Cup, not including the drivers (coxswains).

How many crews of eight rowers can be made?

Ask students to work out the answer in their head if they can and record their strategy on paper. Give the students an appropriate amount of thinking time. Then ask them to share their solutions with their learning partner. The following are possible responses:

Place value partitioning (chunking): 
184 ÷ 8
I know that 160 ÷ 8 = 20. That is 20 crews.
There are 24 rowers left. 24 ÷ 8 =3

The answer is 20 + 3 = 23.

View how to do this using Video Four.

Factorisation (proportional adjustment): 

Dividing by 8 is like dividing by 2 then 2 then 2 so half 184 is 92 and half 92 is 46 and divide by 2 again leaves me 23 so the answer is 23.

View how to do this using Video Five.

Rounding and compensating: 
If there were 200 rowers, that would be 25 crews because 4 x 25 =100 so 8 x 25 = 200.  184 is 16 rowers less. That is two crews. So, the answer is 25 – 2 = 23.

As different strategies arise ask the students to explain why they chose to solve the problem in that way. Accept all the strategies that are elicited at this stage, recording them to reflect upon later in the unit (perhaps in a modelling book). Watch for repeated subtraction or partial use of multiplication, such as:

10 x 8 = 80

80 + 80 = 160, 20 x 8 = 160

160 + 8 = 168, 168 + 8 = 176, 176 + 8 = 184

So 23 crews can be made

View how to do this using Video Six.

Ask students to reflect on the strategies that have been discussed in the session and evaluate which strategies that they personally need further work on, perhaps using thumb signals - thumbs up - confident and competent with the strategy, thumbs sideways - semi confident, thumbs down - not yet confident. Use this information to plan for your subsequent teaching from the exploring section outlined below.

Exploring

Over the next two to three days, explore the following strategies, making explicit the strategy you are concentrating on as the teacher and the reason for using the selected strategy. For example, In the problem 7 x 29 tidy numbers would be a useful strategy as 29 is close to 30.

The following questions are provided as examples for the promotion of the identified strategies. If the students are not secure with a strategy you may need to make up some of your own questions to address student needs.

The following questions can be used to elicit discussion about the strategy:

  • What tidy number or numbers could you use that are close to one of the factors in the problem?
  • What do you need to do if you tidy up this number? Why?
  • Why is this strategy useful for this problem?
  • What knowledge helps you to solve a problem like this?

Place Value Partitioning (Multiplication) 
Nick has $54, but he needs 7 times this amount to buy the new mountain bike he wants. How much money does the bike cost?
The place value strategy involves multiplying the ones, and tens separately then combining the partial products. This strategy applies the distributive property of multiplication, as 54 is distributed into 50 + 4.  In the above problem the student might say the following:
I multiplied 7 x 50 and got 350, then I multiplied 7 x 4 and got 28. I added 350 and 28 to get 378.

The following questions can be used to elicit discussion about the strategy:

  • How can you use your knowledge of place value to solve this problem?
  • Why is this strategy useful for this problem?

If the students do not seem to understand the partitioning concept, show the problems physically. Some students may find it useful to record and keep track of their thinking. An extension of the place value strategy involves the use of standard written form for multiplication.

Use the following questions for further practice if required:

  • 3 x 135
  • 9 x 66
  • 6 x 132
  • 8 x 79

Place value partitioning (division)
Pisi has an after-school job at the market, bagging pawpaw into bags of 6. If there are 864 pawpaw to be bagged, how many bags can he make?
The place value partitioning strategy for division involves ‘chunking’ known facts and subtracting them from the answer. The long division written form will be familiar to most teachers. In the case above, a student might think:
100 x 6 = 60 so100 bags would be 600. 864 – 600 = 264. That leaves me with 264.

I can take 120 away from that, which is 20 x 6. That leaves 144. If I take another 120 pawpaw away I get 24, which is 4 lots of 6. So, I’ve taken away 100 lots, then 20 then 20, then 4… the answer is 144.

This thinking could be recorded as:

If the students do not seem to understand the partitioning concept, show the problems physically, e.g. using place value blocks. Some students may find it useful to record and keep track of their thinking. An extension of the place value strategy involves the use of standard written form for division.

Use the following questions for further practice if required:

  • 760 ÷ 5
  • 516 ÷ 4
  • 992 ÷ 8
  • 3808 ÷ 7
  • 522 ÷ 3
  • 2505 ÷ 9

Rounding and Compensating (Multiplication) 
The Southern Sting netball fans are going to Christchurch to watch a netball game against the Canterbury Tactix.

Each bus is full, with 48 people in it, and there are 9 buses.

How many Sting fans are heading to Christchurch?

The rounding and compensating strategy involves rounding a number in a question to make it easier to solve. In the above question 48 can be rounded to 50 (by adding 2). The problem then becomes 9 x 50 = 450. In order to compensate for the rounding, two lots of 9 people (18) must be subtracted from the ‘rounded’ equation.

If the students do not seem to understand the tidy numbers concept, use place value equipment or a large dotty array to show the problems physically. Some students may find it useful to record and keep track of their thinking. Recording might look like this:

9 x 48 = 9 x 50 – 9 x 24                              9 x 40 = 10 x 48 – 1 x 48
= 450 – 18                                                     = 480 - 48
= 432                                                              = 432
 

View how to do this using Videos Seven and Eight.

Use the following questions for further practice if required, still using the same bus context:

  • 7 x 29
  • 6 x 38
  • 5 x 57
  • 3 x 69
  • 4 x 97
  • 8 x 36

Note that the problems posed here are using a tidying up strategy rather than tidying down. If one of the factors is just over a tidy number (such as 42) then standard place value partitioning tends to be a more useful strategy.

Rounding and compensating (Division) 

Sarah uses nine bus tickets every week to travel around town. She wins 162 tickets in a radio competition. How many weeks will the tickets last her?
Rounding and compensating for division involves finding a number that is close to the dividend (starting amount) and working from that number to find an answer. For the question above, a student might say:

I know that 20 multiplied by 9 equals 180. 162 is 18 less than 180, that’s 2 x 9.

The tickets would last her 20 – 2 = 18 weeks.

If the students do not seem to understand the rounding and compensating concept, use place value materials, or a large dotty array, to represent the problems physically. Students may find it useful to record and keep track of their thinking, especially if they partially divide the dividend at first.

View how to do this using Video Seven.

Use the same context of buses to pose problems where rounding and compensating is a sensible strategy.

  • 343 ÷ 7
  • 224 ÷ 8
  • 597 ÷ 3
  • 392 ÷ 4
  • 1764 ÷ 18

Proportional Adjustment (Factorisation) 
At the Kapa Haka festival there are 32 schools with 25 students in each choir, how many students are there altogether in the choirs?
Proportional adjustment involves using knowledge of factors and multiples to create easier equations that have the same answer. Factors are proportionally adjusted to make one (or both) factors easier to work from. In the above problem the factors could be adjusted as follows:
32 x 25 arrows devide by 4 = 8 x 100
Or, using doubling and halving:
(32 devided by 2 = 16) x (25 x 2 = 50)

The following questions can be used to elicit discussion about the strategy:

  • What could you multiply one of these numbers by to make it easier to work with?
  • What would you then need to do to the other number?
  • Why is this strategy useful for this problem?
  • What knowledge helps you to solve a problem like this?

If the students do not seem to understand the proportional adjustment concept, use a large dotty array to show the problems physically. Some students may find it useful to record and keep track of their thinking.

View how to do this using Video Eight.

Use the following questions for further practice if required:

  • 25 x 200 (multiply and divide by five)
  • 27 x 3 (thirding and trebling)
  • 33 x 18 (thirding and trebling)
  • 44 x 25 (multiply and divide by four)
  • 24 x 125 (multiply and divide by eight)

Proportional Adjustment (Division) 
Tim delivered the same number of pamphlets each month.

In his first six months the delivers a total of 912 pamphlets?

How many did he deliver each month?

In division, proportional adjustment involves changing both numbers in the equation by the same factor. Therefore, the numbers used to proportionally adjust the problem must be factors of both numbers in the equation (dividend and divisor). For example:
If I divide the 912 by 3 and 6 by 3, my equation becomes 304 ÷ 2 which has the same answer. Half of 304 is 152. So, Tim delivers 152 pamphlets.

The following questions can be used to elicit discussion about the strategy:

  • What is a common factor of both numbers that could be used to make the problem easier?
  • Why is this strategy useful for this problem?
  • What knowledge helps you to solve a problem like this?

If the students do not seem to understand the proportional adjustment concept, use equipment to show the problems physically. Some students may find it useful to record and keep track of their thinking.

Use the following questions for further practice if required:

  • 636 ÷ 12 = 212 ÷ 4 = 106 ÷ 2 = 53
  • 480 ÷ 15 = 96 ÷ 3 = 32
  • 1962 ÷ 18 = 981 ÷ 9 = 109
  • 1498 ÷ 14 = 749 ÷ 7 = 107
  • 1728 ÷ 16 = 864 ÷ 8 = 108

Factorisation (Multiplication and Division) 
Stephanie has 486 marbles to share evenly amongst eighteen of her friends. How many marbles will each person get?

The factorisation strategy involves using factors to simplify the problem. In this instance eighteen can be factorised as 2 x 3 x 3. This means dividing by two, then three, then three has the same net effect as dividing by 18. Likewise, multiplying by two, then three, then three has the same net effect as multiplying by 18. In applying factorisation to the above problem, a student might think:

18 is the same as 2 x 3 x 3. So I halve 486, then third, then third. If I divide 486 by 2 I get 243. 243 divided by 3 is 81. 81 divided by 3 equals 27. The answer is 27.

The following questions can be used to elicit discussion about the strategy:

  • How can you use your knowledge of factors to solve this problem?
  • Why is a factorisation strategy useful for this problem?

If the students do not understand the factorisation concept, show the problems physically. Some students may find it useful to record and keep track of their thinking.

Use the following questions for further practice if required:

  • 532 ÷ 8 (÷2, ÷2, ÷2)
  • 348 ÷ 12 (÷2, ÷2, ÷3)
  • 4320 ÷ 27 (÷3, ÷3, ÷3)
  • 135 x 12 (x2, x2, x3)
  • 43 x 8 (x2, x2, x2)
  • 27 x 16 (x2, x2, x2, x2)

Each day follow a similar lesson structure to the introductory session, with students sharing their solutions to the initial questions and discuss why these questions lend themselves to the strategy being explicitly taught. Conclude each session by having students make some statements about when this strategy would be useful and why (e.g. "place value is useful when there is limited renaming required" or "factorisation is useful when one of the factors is able to be renamed as a series of smaller factors"). It is important to record these key ideas as they will be used for reflection at the end of the unit.

Reflecting

As a conclusion to the weeks work, give the students the following five problems to solve asking them to predict which strategy they think will be useful for each problem and why they think this is the most useful strategy before they solve them. After they have solved the problems engage in discussion about the effectiveness of their selected strategies for the problems.

There may be problems for which two or more multiplication and division strategies are equally efficient. However, using additive strategies with these problems will not be efficient.

Problems for discussion (more than one strategy might be suitable for these) 

  • 48 x 50 (proportional adjustment)
  • 559 ÷ 13 (place value partitioning)
  • 29 x 16 (rounding and compensating)
  • 1926 ÷ 18 (proportional adjustment)
  • 212 x 11 (place value partitioning)
  • 704 ÷ 8 (factorisation)
  • 153 ÷ 17 (rounding and compensating)
  • 421 x 8 (factorisation)

Ask the students to create problems for a partner where one of the strategies covered in this unit is the most useful.

Conclude the unit by showing the students the questions asked in the initial session again and discuss whether they would solve them in a different way now, why or why not. Review the modelling book or record of statements or generalisations about the strategies and make any changes.

Transformations

Purpose

This unit explores important ideas about reflective and rotational symmetry.

Achievement Objectives
GM4-8: Use the invariant properties of figures and objects under transformations (reflection, rotation, translation, or enlargement).
Specific Learning Outcomes
  • Recognise when one figure is a reflection of another using variant properties such as perpendicular distance from the mirror line, equalities of lengths, areas and angles, and opposite orientation in relation to the mirror line.
  • Recognise the rotational symmetry of a figure including identifying the centre of rotation and the order and angle of rotation.
  • Create symmetrical patterns using the properties of translation, reflection, and rotation.
Description of Mathematics

Both reflection and rotation are isometries which means that measures remain the same. The lengths, angles and areas of 2-dimensional shapes and their images under reflection and rotation are the same. For example, in the image below a polygon is reflected in the y-axis. The lengths AB and A’B’ are the same, as are <CDE and <C’D’E’ and the areas of the polygon and its image (8½ square units).

The only changes from shape to image in reflection and rotation are to do with orientation. In reflection the shape and image face opposite directions relative to the line of reflective symmetry (mirror line). In rotations orientation depends on the angle of rotation.

Invariance of length, angle and area are insufficient to locate the image of a shape under refection and rotation. In reflection the location of points and their image are the same perpendicular (90◦) distance to the mirror line. B and B’ are both 3 units of length away from the y-axis at right angles.

In rotation points on the shape and on the image are the same distance to the centre of rotation, the point through which the shape was rotated. In this example the distances of both A and A’ to the centre of rotation (0,0) are equal. Note that the angle formed by the distance rays is 90◦ the angle of rotation anti-clockwise.

Specific Teaching Points

Research by Ramful and Lowrie (2015) shows that students tend to rely on their spatial thinking rather than analytically considering whether or not a shape is the image of another under refection or rotation. Spatial thinking involves capturing an image of a shape and transforming it mentally. Spatial imagery is important and is extremely useful in a wide variety of fields, such as engineering, design, architecture, building and surgery.

However, advancing spatial reasoning requires students to connect their spatial imagery and analytic thinking. Analytic thinking involves considering the properties of shapes under transformation. Knowing and applying the invariant properties of reflection and rotation allows students to solve more complex problems than those accessible only be visualisation.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students, or by varying the task requirements. Ways to support students include:

  • pairing students so that they can support each other. This is particularly important as they work on computer applications
  • providing pre-made cardboard shapes so students can physically act out reflections and rotations
  • restricting the complexity of the shapes students are asked to reflect or rotate
  • using computer technology to encourage risk taking and experimentation.

This unit is focuses on reflection and rotations. These symmetries are common in everyday figures. Use examples from real life that are appropriate to the interests and cultural identities of your students. Traditional Māori and Pasifika designs such as kōwhaiwhai, and tapa cloth, use reflection and rotation. Logos of well-known brands, tiles, wallpaper patterns, and natural shapes in the environment, such as reflections in lakes, provide other possible contexts.

Required Resource Materials
  • Attached files as listed at bottom of unit
  • Paper, card (waste), pens
  • Access to computers
  • Copymasters One, Two, Three, Four, Five and Six
Activity

Prior Experience

It is anticipated that students will have prior experience of reflecting and rotating shapes without necessarily attending to invariant properties. They may have experience of creating patterns such as tessellations and kowhaiwhai by use of reflection and rotation.

Session One

In this session students explore the invariant properties of reflection.

  1. Search for a youtube clip of “Harry Worth Window”. You will find a short introduction to a comedy show where Harry, an English comedian, uses a reflection to create an illusion that he is split jumping in the air.
  2. Ask, “How does the trick work?”
    The camera only sees one real half of Harry. The other half is the image of his real half after reflection. A student might physically model how the effect works.
  3. Use Copymaster 1 for students to draw stages of Harry’s ‘jump’ as follows.
  4. If your students have access to appropriate software they can use that software to create an interactive ‘jumping Harry’. Video One A gives instructions of how to create an interactive model on Geogebra. 
  5. Ask the students to work in small groups to discuss what features of the real and image halves of Harry for the effect to work. Invariant properties such as conservation of length, angle, area should arise. It is also important that students realise that image has opposite orientation to the real half. The image holds up it’s right hand while the real half holds up it’s left hand.
  6. Pay particular attention to what becomes of the key points (label them) when they are reflected to form the image. This may need to be done systematically. For example:

     Real Point

     A (2,1)

     B (1,6)

     C (0.5, 8)

     

     

     

     Image Point 

     A’ (-2, 1)

     B’ (-1, 6)

     C’ (-0.5, 8)

     

     

     

     

  7. Expect students to notice that reflection in the y-axis maps a co-ordinate (x, y) onto (-x, y). Software makes this more obvious as the co-ordinates of key points are usually available visually. Note that the distance from a point on the real half to the mirror line is the same as the matching distance of the image to the mirror line. That distance is measured at right angles to the mirror line. This property explains why any point, for example (1, 6), will have an image (-1,6) since 1 and -1 are the same distance from the y-axis.
  8. Use Video One B to introduce the game of Symmetry Tennis. The example shown is a more challenging form of the game since the mirror line is at an angle, rather than being horizontal or vertical. Provide students with Copymaster 2 so they can play the game in pairs.
    Look for the following:
    • Do students attend to the invariant properties, particularly the perpendicular distance of a point to the mirror line?
    • Do they attend to orientation?
    • Do they play strategically by creating shapes that are difficult to locate and position?

Session Two

In this session the theme of symmetry is continued with focus on the internal symmetry of a shape. Students will need a digital picture of their face looking straight onto the camera with their eyes level.

  1. Start with Video Two A. The first frame shows the face of a young man. Ask, “How could we check how symmetrical this person’s face is?” Pause the video and Invite ideas from the students. Encourage them to apply the invariant properties of reflection that they discovered in the previous lesson. For example, the pupils of his eyes should be the same distance from his mid-line and at the same height. The angle formed by his cheeks should be the same on both sides. The areas of his ears should be the same. You may chose to let students investigate further with their own idea or continue with the video.
  2. Watch the video further to see how overlaying a co-ordinate plane can support closer analysis using key points. Once the students understand how key points can be located discuss how those points can be used to get some idea of how symmetric the face is. Ideas that might arise are:
    • Co-ordinates will need to take on decimal values to be accurate enough.
    • Points should match with the x value of the ordered pair changing signs, e.g. (2.7, 3.3) should map to (-2.7, 3.3).
    • Some way of measuring variation will need to be developed.
  3. Put the students into small groups to establish some idea of how symmetric the young man’s face is. Provide the students with a copy of Copymaster 3 for their group to work on. Let the students work collaboratively for a suitable time before bringing the class together to discuss their thoughts. Student might invent some measure of symmetry. The measure might be absolute, such as difference in the numbers in the ordered pairs, e.g. eyebrows – (2.2, 5.9) is only 0.2 different to (-2.3, 5.8). Others might create a proportional measure, e.g. -2.2/-2.3 = 0.96 = 96%. Useful questions to ask are “Should we expect that the ordered pairs will match exactly? How do we allow for error in our measurement? What will be appropriate variation?
  4. There are several possible options for extending the activity:
    • Let students investigate the extent to which their own face is symmetric. It is worth reminding them that even people perceived as glamorous are non-symmetric. You can easily find examples on the internet.
    • Let the students investigate what they would look like if two halves of the right side of their face were merged. Do the same thing with their left side. Video Two B shows how to do this.
    • Develop of symmetry measure collectively and apply it to their own face or that of a well-known celebrity or sports star.

Session Three

In this session students extend their concept of symmetry to include rotational symmetry, looking for invariant properties and testing to see if a shape has internal rotational symmetry.

  1. Look at Video Three A which shows a design from traditional Celtic knot design. Pause the video after the introduction to ask the students what symmetry the shape has. It is likely that students will initially focus on refection. Remind them how the shape might be tested for reflection using invariant properties. The knot does not have reflective symmetry. This is particularly obvious at points where one thread goes over or under another.
  2. Play the video to illustrate how the figure maps onto itself by rotation about the centre. The order of rotational symmetry for the knot is four, meaning it maps onto itself four times in a full 360◦ rotation. The angle of rotation is 90◦.
  3. Copymaster 4 has other Celtic knots. Ask the students to establish the symmetry of each knot. Can they:
    • recognise when a shape has or has not got reflective symmetry?
    • recognise when a shape has or has not got rotational symmetry?
    • establish the angle and order for shapes with rotational symmetry?
  4. You may like to look at this interactive worksheet about the rotational symmetry of a snowflake and regular octagon. Before dragging the sliders ask your students to anticipate which angle will map the shape onto itself.
    https://www.geogebra.org/m/PhvUvyqz (symmetry of a snowflake and octagon)
  5. Next play Video Three B which gives instructions for creating a slider interaction in GeoGebra. Be aware that some drop down menus do not show in the video but you are ‘talked through’ how to use them. You may decide to show the video in small sections, so students can create their own interactions as they view. Alternatively, if technology is not accessible, use Copymaster 5 so students can do the same activity by manually drawing a polygon with integral key points.
  6. After students have created their own interactions pose the following investigation:
    Suppose you have a polygon that has some key points. These key points have co-ordinates such as A = (1,2), B = (3,5), C = (6,6) etc.
    Rotate the polygon 180◦. What is the connection between the co-ordinates of each key point and the co-ordinates of the image of those points, A’, B’, C’, etc.
    Can you generalise what happens to any key point under a 180◦ rotation about the origin?
    Try rotations of 90◦ and 270◦.
    Can you generalise about the image of any key points under these rotations?
  7. Discuss students' answers to the questions. In general, any key point (x,y) becomes (-x,-y) under a 180◦ rotation, (x, y) becomes (-y, x) under a 90 rotation anti-clockwise, and (x, y) becomes (y, -x) under a 90 rotation clockwise.
  8. Finalise the lesson with the questions on Copymaster 6. Check to see if your students can identify the invariant properties of a shape under rotation.

Session Four

In this session students engage in an investigation that applies reflective and rotational symmetry. Students will work in small groups. Each group will need a set of paper squares which are readily available at stationery and dollar shops. They will also need a compass.

  1. Show the students how to create a set of Quarcibits as follows. This investigation was originally created by Dr Jill Brown (Australian Catholic University).
  2. The two different Quarcibits can be joined by joining matching edges. So…
  3. The first part of the investigation is to find out how many different shapes can be made by joining two Quarcibits. Remind the students that two quarter circles and two left over pieces could be joined.
  4. Give your students time to find all the possible shapes. A shape is the filled silhouette so the location of the join is not important. Ask students to come up with a convincing argument that they have found all the possibilities. Expect a systematic strategy such as:
    • Classify the shapes by combinations, i.e. two quarter circles, quarter circle and left over, two left overs.
    • For each ‘family’ anchor one Quarcibit then systematically consider the possible positions of the other Quarcibit.
    • Test the shapes that are created for uniqueness. If shape maps onto another by reflection or rotation it is not unique.
  5. There are seven di-Quarcibits possible as shown below, organised by family of meeting Quarcibits:

    Giving the shapes names is helpful in identifying them, especially if the names relate to real world shapes, such as ‘hedgehog’ and ‘lightning bolt’.
  6. Next play Video Four A to show your students how the di-Quarcibit shapes might be organised in a Carroll Diagram by symmetry. Ask your students to create the two-way table and classify their di-Quarcibits. Look for:
    • Do students work with both classifications simultaneously to locate the shape in the correct cell?
    • Can they identify if a shape has symmetry and the nature of that symmetry (lines, order, angle of rotation)
  7. To extend the investigation further challenge your students to find all the tri-quarcibits. There are 21 possible shapes and most lack symmetry – Why?

Solving linear equations

Purpose

This unit teaches students to identify linear relationships and solve linear equations in context.

Achievement Objectives
NA4-7: Form and solve simple linear equations.
NA4-9: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.
Specific Learning Outcomes
  • Identify and find values for variables in context.
  • Identify linear relationships in context.
  • Represent linear relationships using tables, graphs and simple linear equations.
  • Draw strip diagrams to represent linear equations.
  • Solve simple linear equations and interpret the answers in context.
Description of Mathematics

Algebra started with the need to solve problems. Al Khwarizmi, a Persian mathematician, was arguably the first person to represent linear and quadratic problems in symbolic form and solved the problems by processes of ‘restoration’, i.e. equivalent operations that conserved equality. In fact, the word for algebra comes from the Arabic word for restoration.

It is fitting then that modern approaches to algebra focus on the thinking that underpins the symbolic systems. Algebraic thinking is concerned with generalisation. Letters, words, tables, graphs, networks, etc. are cultural tools that enable us to represent, then think with, those generalisations. With representational tools we are capable of ‘amplified cognition’ in that we can anticipate results that would never be possible if we relied solely on the physical environment, and on our limited capacity to process ideas just mentally.

Generalisation begins with noticing patterns and structures. A pattern is a consistency, that is something that occurs in a predictable way. It is the ‘what’ of algebraic thinking. Structure is about the organisation of patterns. It is the ‘how’ and sometimes the ‘why’ of generalisation. From noticing pattern and structure, we develop properties. For example, early counting involves pattern and structure. The ‘fourness’ of a collection comes from noticing sameness among collections of four, irrespective of the size, colour, texture, etc. of the objects. Structure of counting involves ideas like the order of counting the objects doesn’t matter.

Specific Teaching Points

In upper primary school, learning experiences for algebraic thinking typically begin with patterns. Usually these patterns are spatial and may be connected to some meaningful life context, though number patterns are also rich in opportunity. Patterns involve variables, that is features, some of which can be quantified. For example, consider this simple spatial pattern.

Among the variables we might discern that the ‘tower’ has height and each ‘tower’ is made of some number of squares. Height and number of cubes may not be the only variables, just those we notice. Variables change, that is height varies and so does the number of squares in the ‘tower’. We might try to find a relation between the variables, describe and represent that relation, and use it to predict how the pattern grows beyond what we can see. Then we are thinking with the properties and representations in a sophisticated way.

On the way it is likely we will need to organise the data from the pattern systematically. A table of values is a productive generic strategy, so we represent the pattern like this:

The danger in moving to an organised numeric strategy like a table too early is that it may negate what we can ‘see’ in the pattern visually. Noticing and reasoning may be inductive, that is tied to the incremental change of the figures. For example:

Noticing and reasoning can also be abductive, that is based on the structure of one example.

Noticing and reasoning can be deductive, that is based on making assumptions about structure and reasoning with the assumptions. For example, we might assume that the tower is composed of an array of something multiplied by three plus two.

From the assumptions we might deduce the appearance of towers much further on in the sequence, e.g. A tower 100 high will contain 2 + 99 x 3 squares. Ways of ‘seeing’ the pattern are manifest in relations within the table of values. For example, inductive thinking leads to seeing the values in the bottom row increasing by three each time. Abductive reasoning might support seeing this relation in the table:

Representing the relation as an algebraic equation involves two important and connected types of knowledge, related to the language conventions (semiotics), and to the nature of variables. We might write s = 3h – 1, or s = 3(h - 1) + 2, or s = 2h + (h – 1), depending on what we notice. The equations are meaningless to anyone else unless we clearly define what the variables, s and h, represent. Note that both and s refer to quantities that vary and are not fixed objects, such as houses or towers. Quantities are a combination of count and measurement unit. In this case h expresses unit lengths in height, and s refers to an area of squares. 3h means h multiplied by three, not thirty-something, and 3(h - 1) means that one is subtracted from h before the multiplication by three occurs. Working with variables requires acceptance of lack of closure, that is thinking with an object (h in this case) without specifically knowing what it is. For example, knowing that 3(h – 1) = 3h – 3 is true, irrespective of whatever the value of h, is itself a generalisation. The equals sign represents a statement of ‘transitive balance’ meaning that the balance is conserved if equivalent operations are performed on both sides of the equation. Knowledge of which operations conserve equality and those which disrupt it are important generalisations about the properties of numbers under those operations, e.g. distributive property of multiplication.

This unit specifically deals with relations that are linear. The first sign of linearity is that there is constant difference in the increase or decrease of one variable, as the value of the other increases by one. In the table above the number of squares increases by three as height increases by one.

Note that this graph shows a relation, not a function, since the values of variables are discrete, not continuous. There are some important connections between features of the algebraic equation, the table and the graph of a linear relation. Constant difference is represented by the co-efficient of the independent variable (s = 3h -1 in this case), differences of three in the bottom values of the table, and a slope of three (change in s for every unit change in h). The constant in the equation (- 1) is reflected in the table by a need to adjust the value of 3h by subtracting one to get the value of s, and reflected in the graph as a downward translation (shift) of the graph for s = 3h by one unit. This results in the intercept of the graph with the s axis being (0, -1), not the origin (0, 0).

Simple linear equations occur when the value of one variable in a relation or function is set and the other must be found. For example, with the tower problem this problem might be posed; “A tower in the pattern has 98 squares. How high is the tower?” Depending on the equation used to represent the relation, this problem can be expressed as 3h – 1 = 98, 3(h – 1) + 2 = 98 or 2h + (h – 1) = 98. Linear equations with the variable on both sides occur when two conditions are equalised. An example might be, “Both Lilly and Todd look at the same tower. Lilly notices that the number of squares in the tower is three times the height less one. Todd notices that the number of squares is two times the height plus 18. How tall is the tower?” This problem can be written as 3h – 1 = 2h + 18.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

  • Use a linear model to develop the concepts of variable, as represented by a letter, and constant, as represented by a number.
  • Develop the use of tables and graphs to organise data from a situation and look for patterns.
  • Work with digital tools, including spreadsheets and graphing application, to develop rule seeking and equation forming.
  • Explicitly teach the conventions of the same operation applied to both sides of an equation. Discuss the concept of equality as ‘transitive balance’ and ask students to record steps systematically.
  • Use the four progressive versions of Visual Algebra to differentiate the challenge for students.

The contexts for this unit include Moa, working for money, ropes for gymnastics, swimming, stacking supermarket trolleys, and video games. Those contexts will appeal to most students. Use other contexts relevant to your students, in which there is a constant difference between consecutive terms (linear relationships). For example, the coach takes his car that holds three extra players. All the other players must travel on vans, seven players to a van.  

Required Resource Materials
Activity

Prior Experience

It is anticipated that students at Level 4 understand, and are proficient with, multiplicative thinking. However, the tasks in this unit are also accessible for students whose preference is additive thinking. In fact, the experiences may prompt a move towards multiplicative thinking.

Session One: Maia the Moa

In this session students are shown a spatial growth pattern for a moa made from square tiles. As Maia the moa ages she grows in her legs, body and neck while her feet and head remain constant. Session One is driven using PowerPoint One. The approach is to structure one example of the pattern then transfer that structure to other members of the pattern.

  1. Show the students Slide One. Aim to identify features of the pattern that might become variables. Ask: What do you notice about this figure?
    Students might notice different features such as colour, height, width, age, total number of squares, etc.
  2. Ask: Is there an easy way to count the number of squares that Maia is made of?
  3. Give students a while to structure their counting then ask them to share their method with others. Building a model of Maia at age three years with connecting cubes allows students to experiment with ways to partition the model. Encourage them to express their counting method as an expression. Use these videos to show examples of how to do this, but only if needed:
  4. Ask students to apply their counting structures to Maia at age two years (Slide Two). Ask them to record expressions for their counting strategy and compare them to what they recorded for year four.
  5. Ask: What changes and what stays the same in your expressions?
    For example, from Casey’s method these two expressions emerge:
    4 + 2 x 3 + 2 (Age two)                   6 + 2 x 5 + 2 (Age four)
    The ‘+ 2’ is constant and ‘2 x’ is present in both expressions. The other numbers vary.
  6. Ask: What will your expression for Maia at age three years look like? Write the expression then check it by drawing a picture of Maia at age three (See Slide 3).
  7. Ask students to show where the parts of their expressions come from in the picture. For Casey’s method the expression is 5 + 2 x 4 + 2. Slide 4 shows how parts of the diagram can be linked to parts of the expression. Look at the strategies of the students.
    Are their strategies based on induction? That is sequential processing. For example, 4 , ? , 6, so ? = 5, and 2 x 3, 2 x ?, 2 x 5, so ? = 4.
  8. Are their strategies based on deduction? That is reasoning about the structure of any term. For example, the first number is two more than the age, and the multiplier of two is one more than the age. So, for y = 3 Casey’s expression is 5 + 2 x 4 + 2.
  9. Pose this problem for students to explore individually or in small co-operative groups:
    Imagine that Maia celebrates her twentieth birthday.
    How many squares will she be made of?
    Find a way to predict the number of squares that Maia is made of for any age in years?
  10. Allow students plenty of time to explore the problem. Look for the following:
    • Do the students record the data systematically? For example, if they draw Maia at age five years. Are their structural counting methods consistent? Is their recording in sequence?
    • Do students use inductive methods? For example, Maia increases by three squares each year.
    • Do students use deductive methods? For example, applying Casey’s method Maia should be (20 + 2) + 2 x (20 + 1) + 2 on her twentieth birthday.
    • How do students express their general rules? Do they use words?, e.g. “I take the age and add two to it to get the first number..” or do they attempt to symbolise their rules, e.g. Next number = number before + 3.
  11. Bring the class together to discuss their methods with emphasis on the points above. Acknowledge the legitimacy of inductive methods but also highlight the power of deductive methods. Use questions like, “Which strategy would be better for finding out about Maia at 100 years of age?”

Session Two

This session builds on the Maia, the moa, pattern to represent the relation between age and number of squares using a table, a graph and an equation. Features of these representations are connected through looking at the effect of changing the original spatial pattern with focussed variation.

  1. Open Excel or a similar spreadsheet program and create a blank workbook. You may need to have Slide 3 of PowerPoint One available for source data. Ask one of the students to set up a table like this:
  2. Ask students what they notice in the table.
    Some may notice missing values in the Age column, particularly the ages 0, and 1. Others may notice that the number of squares are all multiples of three. They may express this idea inductively, “The number of squares goes up by three.”
    How can we continue the table to get more values?
  3. Induction can be used to ‘fill down’ the values in both columns but deductive rules across the columns are more sophisticated. Video 2A shows how to create values by filling down. Video 2B is about using formulae across the columns. The videos can be stopped at any point for discussion. Video 2B goes straight to the most efficient rule but students could enter the rules they developed in Lesson One.
  4. Ask: Can you use Excel to show that your rule from yesterday works?
  5. Next a graph is created from the table of values. Video 2C shows how to do this. Ask the students to create their own graph of Maia’s growth patterns and record some features that they notice.
    Why are the points in a line? (This tell us that the relation is linear)
    How steep is the line?
    Note (0, 6) represents Maia’s situation upon hatching.
    Where does it cross the s axis?
    Why does it cross there?
  6. PowerPoint Two offers three scenarios in which Maia’s shape is changed in some way. The reason for doing this is to connect features of the table and graph with the spatial pattern. For each scenario students may need to draw the progression of each pattern back until Maia hatches. That will lead a table of values that can be graphed. Video 2D shows what happens when the original Maia growth pattern is altered by a constant, - 1 for losing her foot and + 2 for gaining a backpack. Video 2E and Video 2F show the effect of changing the co-efficient (multiplier) of a, in that the slope of the graph alters from three to four. Copymaster 1 provides printable versions and the start of a table for each.

Session Three

  1. Remind the students of the rule that was entered into Excel to create the pattern in the Number of squares column for Maia’s original growth pattern (e.g. =(A2+2)*3).
    What does A# represent? (Maia’s age in years, a). So instead of A# we could write = (a + 2) x 3 or = 3(a + 2).
    What does this expression tell us? (The number of squares Maia is made up of). So we could write s = 3(a + 2).
  2. Show the students PowerPoint Three which shows how linear equations can be represented using a length model. Work through the slides.
    Do the students observe that a is free to take up different values? a is a variable. The twos remain equal in length as the value of a changes. So, +2 is a constant.
    Pose this problem to the students.
  3. Maia is made up of 144 squares. How old is she, in years?
    This situation constrains s to 144 so a linear equation is created which might be expressed as 3(a + 2) = 144 or in other forms, dependent on the structure of the rule. For example, Katia’s method would yield 3a + 6 = 144. Student may need access to a picture of Maia’s growth pattern, e.g. Slide 3 of PowerPoint One.
  4. Look to see whether the students use deductive reasoning or whether they are reliant on inductive methods.
    For example, inductive methods might involve creating a table of values and extending it until the matching value of a is found. Spreadsheets make inductive methods easy to implement. A sign of reliance on additive methods would be repeated adding of three to find next values of s.
    Deductive methods involve applying inverse operations to rules. For example, “I divided 144 by three to get 48, so the age plus two must equal 48.”
  5. After a suitable time gather the class to discuss their strategies. Highlight the efficiency of deductive rules, which are sometimes referred to as function or direct rules, compared to lengthy inductive rules, which are sometimes referred to as recursive. Slides 5 and 6 show one way to solve the problem of Maia’s age when she is made of 144 squares.
  6. The 144 squares problem shows how solving linear equations can lead to solutions efficiently. Play this video which introduces how to use the simplest version of the Visual Linear Algebra learning object. Allow students plenty of time to explore the object.

Session Four

In this session students investigate linear equations where the variable is present on both sides.

  1. Begin with a reminder of how to solve linear equations in their simplest form by looking at the structural similarity of possible rules for Maia’s growth pattern. PowerPoint Four gives two possible rules attributed to hypothetical students. The rules may be alike some that the students created in Session One and Two. Slide Four shows the lengths rearranged end on end.
  2. Ask: Why do these rules give the same total for any value of a?
    Do students recognise that both rules can be rearranged to give 3a + 6 which is Katia’s rule?
  3. Possibly link the algebraic manipulation that matches the lengths in the diagram is students show interest. For example:
    (Leah’s rule) 3 (a + 1) + 3 = 3a + 3 + 3
    = 3a + 6 (Katia’s rule)
  4. Ask the students to use Katia’s rule to solve this problem:
    Maia the moa is made of 222 squares. How old is Maia?
    Do students apply inverse operations to both sides of the equation, 3a + 6 = 222, to find the solution?
  5. Pose this problem:
    Ken and Katia are looking at the same picture of Maia.
    Katia says that the number of squares equals three times Maia’s age plus six.
    Ken says that the number of squares equals four times Maia’s age minus 18.
    They are both correct. How old is Maia?
  6. Let the students work in small groups to solve the problem. Look for the following:
    • Do they build up a table of value inductively to find a value for a that meets both conditions?
    • Do they try values of a and ‘close in’ on the solution?
    • Do they use their knowledge of equations to solve the problem?
  7. Bring the class together to share their solution methods. Trial and improvement strategies can be very efficient in solving these types of problems, especially if the initial attempts are based on reasonable estimation. For example, setting a = 30 gives Ken’s number of squares at 102 and Katia’s at 96. So, is 30 too big or too small?
    An equation based solution looks like:
    3a + 6 = 4a – 18
    3a + 24 = 4a (adding 18)
    24 = a (subtracting 3a)
    Note that there are many possible first moves.
  8. Introduce the second learning object in the Visual Linear Algebra collection using this video.  Allow students plenty of time to explore the tool.

Session Five

This session is intended as an opportunity for students to practice applying their understanding of linear relations and their techniques for solving linear equations.

Provide the students with copies of Copymaster 2 and encourage them to solve the problems in co-operative groups.

Addition, subtraction and equivalent fractions

Purpose

The purpose of this series of lessons is to develop understanding of equivalent fractions and the operations of addition and subtraction with fractions.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-5: Know fractions and percentages in everyday use.
Specific Learning Outcomes
  • Add and subtract fractions with like denominators.
  • Explore and record equivalent fractions for simple fractions in everyday use.
  • Recognise that equivalent fractions occupy the same place on the number line.
  • Recognise and apply the multiplicative relationship between simple equivalent fractions.
  • Understand that fractions can have an infinite number of names.
  • Understand the inverse nature of multiplication and division and how this knowledge can be used to simplify fractions.
  • Apply knowledge of equivalent fractions to solving problems which involve comparing, adding and subtracting fractions with different denominators.
Description of Mathematics

In this unit students learn to find equivalent fractions and apply equivalence when adding and subtracting fractions with different denominators. A fundamental idea about addition and subtraction is that the units that are combined, separated, or compared are of the same size. In a simple addition problem such as 6 + 7 = 13 it is assumed that 6, 7, and 13 refer to the same units, such as apples, or centimetres.

Addition and subtraction of fractions involves greater complexity than with whole numbers.

For example, 3/4 + 2/5 = ? involves two fractions comprised of different units. Three quarters refer to three units that are one quarter in size, while two fifths refer to two units that are one fifth in size. Since quarters and fifths are different sized units they cannot be added. Note that the denominator tells the size of units that a fraction is comprised of.

Renaming the fractions in units of the same size involves creating a common denominator. A common denominator can be any common multiple of the denominators of the fractions being added or subtracted. For example, to solve 4/5 – 2/3 = ? the common denominator of 3 and 5 could be 15, 30, 45, etc. For simplicity, the least common multiple, 15, is commonly used. Partitioning fifths into three equal parts creates fifteenths, so 4/5 = 12/15. Partitioning thirds into five equal parts creates fifteenths, so 2/3 = 10/15. Note that renaming a fraction in equivalent form is like multiplying by one, in fraction form. 4/5 x 3/3 = 12/15 and 2/3 x 5/5 = 10/15. 3/3 and 5/5 are names for one.

Once 4/5 and 2/3 are renamed as equivalent fractions with the same denominator the units are of equal size and can be separated or compared. So, 4/5 – 2/3 = 12/15 – 10/15 = 2/15.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students, by varying the task requirements. Ways to support students include:

  • continued work on multiplication and division basic facts through diagnosis and practice of missing facts
  • consistent use of concrete materials, such as paper strips or sheets, fraction strips or circles, with an emphasis on students anticipating the results of actions
  • explicit modelling of actions on the materials with appropriate questioning about the fractions involved and the size of one
  • linking words and materials to symbols, so that students connect fraction symbols to quantity
  • use of a variety of situations to which addition and subtraction are applied. The situations include joining, separating, and comparing quantities.

Tasks can be varied in many ways including:

  • beginning with fractions that occur through symmetric partitioning of one, i.e. halves, quarters, eighths, etc., before using more difficult partitions such as thirds, fifths, and sevenths
  • working at first with addition and subtraction problems where the equal partitioning of units is easy to enact and visualise, e.g. halves, thirds, quarters
  • using physical and diagrammatic models before the use of symbols, then using both together
  • collaborative grouping so students can support others. Demanding explanation and justification from groups of students
  • reducing the demands for a product, e.g. diagrams with less reliance on writing equations at first but progressing to fluency with equations
  • using digital technology, such as virtual manipulatives, to facilitate visualisation and risk taking.

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Fractions arise from equal sharing and measuring. Fair shares are a common feature of many cultures, particularly Pasfika communities where effort and reward are communally owned. Sharing food is a cultural universal. In Polynesian cultures, shares given to families from collective food and materials gathering or hunting, are often based on number of family members, so the fractions are often unequal. Measurement contexts are ubiquitous in modern life, due to the use of metric measures. Length, area, mass, volume and capacity occur in everyday settings from allocating water, to creating items from fabric, to calculating loads. Historical measures, such as arm spans, head circumferences, and foot lengths, also provide contexts for use of fractions, arising from the need for accuracy.

Required Resource Materials
  • Unifix cubes
  • Adhesive tape
  • Packs of playing cards
  • Strips of paper made from A3 sized paper
  • Hundreds flip board
  • Copymasters One, Two, Three, Four, Five, and Six.
Activity

Session 1

SLOs:

  • Consolidate understanding of where fractions fit on a number line.
  • Add and subtract fractions with like denominators.
  1. Begin this session by skip counting forwards and backwards in common fractions and mixed numerals.
    For example: “one quarter, two quarters, three quarters, one, one and one quarter, one and two quarters, one and three quarters, two, two and one quarter…”

    Use a set of fractions strips (Copymaster One) to build up a number line model as you count.
    Why is 4/4 called one?
    Why is 6/4 called 1 ½?
     
  2. Ask “How do we write 1 using quarters?” (4/4). How do we write 2 (8/4), 3 (12/4) and 4 (16/4)?
    Write on the class/group chart in words and symbols:
    Four quarters is the same as one: 4/4 = 1,
    Eight quarters is the same as two: 8/4 = 2, etc.
    Highlight the equals sign. Ask students to discuss the meaning of = (is the same as, the amount on one side is equivalent in value to the amount on the other side).
    Record their ideas.

    Equals means the same quantity. So, 8/4 and 2 are two ways to record the same number.
     
  3. Pose and write this problem on the class/group chart. “There is a pizza party at Elijah’s house. These bits of pizzas are left over: 1 1/8 vegetarian, 3/4 meat lovers, 1 5/8 pepperoni, 1/2 Hawaiian, 1 1/4 margarita, and 7/8 seafood. Altogether, how much pizza is left over?”
     
    Ask students to work in pairs to reach a solution. Provide Copymaster Two: Pizza Pieces for students to use if needed. Share the strategies as a class and summarise those strategies on the class chart. Encourage students to think flexibly.
    Are any pairs of fractions easy to add? (e.g. 1 1/8 + 7/8). Why?
    How many eighths of a pizza, altogether, are left over?
    Students might divide the total of 6 ½ pizzas and convert that amount into 52/8. Alternatively, they might change each fraction into eighths and total the number of eighths.
     
  4. Provide each student pair with sticky tape and two strips of paper of A3 length (42cm).
    Make your own fraction number line, starting at zero and finishing at eight, where the right-hand edge is. Tape both strips together to get a good length.

    What number will go on the join?
    Where will you locate 1? 7? 4 ½, 6 3/4?
    Please show all the halves, quarters, and eighths from 0 to 8.
    Watch that students locate all numbers, including fractions, on the marks and not in the spaces.
     
  5. Once students complete their number lines, refer to the fractions in the pizza problem posed in part 3 above.
    Have students begin at zero and add 1 1/8 (the vegetarian pizza left over) on the number line. They might use a peg to mark the total amount of pizza each time, as each fraction is added.

    How can we figure out where a jump of ¾ more will land, adding on the meatlovers leftovers?
    Three quarters more than 1 1/8 equals 1 7/8 then 1 whole pizza can be added to get 2 7/8 pizza. One might be added first then 7/8 and the answer is the same.
    Use your number line to add the fractions of leftovers. Check to see that our original answer is correct.
    Roam the room to see that students add the fractions correctly, renaming ones, halves, quarter, and eighths as needed, and recognising when another one (whole) is created.
     
  6. Pose other problems orally, speaking very slowly; e.g. 2 1/2 + 3/4 + 3/4 + 1 3/4 + 3/4 =
    Ask your students to solve the problem by imaging the number line. Ask students to share their result with a partner. Ensure the recorded fractions are visible for students to see.
    Repeat with other examples involving halves, quarters, and eighths. Use pizza pieces to confirm the sums if necessary.

     
  7. Provide each group with two more strips and a piece of sticky tape. Pose the following challenge:
    Make a new number line starting at zero and finishing at eight. Join the strips like last time.
    This time your number line must show halves, thirds, and sixths.
    How will you divide each length of one into thirds?
    How will divide thirds into sixths?
     
  8. Roam the room as students work.
    Do they locate the whole numbers correctly?
    Is the one third unit made by dividing one into three equal lengths?
    Are students aware that one half of one third equals one sixth?
     
  9. Gather the class and discuss and chart key learning from this session. Focus students’ attention on the need for a common denominator when joining fractions (addition) and that the same quantity may be represented with different fractions.
    You might align the first and second number lines.
    Using our number lines, how many names for 1 ½ can we find?
    1 2/4, 6/4, 1 4/8, 12/8, 1 3/6, 9/6.
    What would happen if we added 3/4 and 5/6? What would the answer be?
    Students should recognise that the answer, 19/12, cannot be expressed as an exact number of quarters or sixths. You might use a fraction strips to show that twelfths are needed to express the answer.

Session Two

SLOs:

  • Recognise that denominators must be the same to add or subtract fractions.
  • Understand the meaning of equivalent.
  • Explore and record equivalent fractions for simple fractions in everyday use.
  • Add and subtract simple fractions that have unlike denominators.
  1. Begin the session by posing a subtraction problem in which the denominators are the same.
    For example: The builder has a 3 1/4 metres length of wood. He cuts off 1 3/4 metres. What length of wood is left?
    Let students calculate an answer then check their responses with fractions strips (Copymaster One). The physical action is removal. Taking away 1 1/4 is straight forward but a unit of one metre needs to be broken up to remove the other 2/4. That leaves 1 2/4 or 1 1/2.
    Review strategies from Session 1. Just like with addition, units of the same denominator are needed for subtraction of fractions.
     
  2. Pose a problem in which the denominators are not easily related.
    For example: The builder has a 4 3/4 metres length of wood. He cuts off 2 1/3 metres. What length of wood is left?
    Ask students work in pairs to attempt the problem. Encourage access to their previous number lines and fraction strips. 
     
  3. Support students using the strip model and scaffolding questions like:
    How might you find the length that remains? (Either add on to 2 1/3 until you reach 4 ¾ or remove 2 1/3 from 4 ¾).
    Estimate the length that remains. (2 metres and a fraction of a metre)
    What fraction is ¾ with 1/3 taken away? What sized pieces might fit? (Twelfths)
     
  4. After an appropriate time gather the class discuss strategies. Model the problem with fraction strips so symbols can be related to quantities.
    Record the problem using an empty number line to capture the removal of parts. 

    What is the problem with 3/4 - 1/3? (Different denominators means different sized pieces)
    Could we rename both 3/4 and 1/3 using a different denominator? Which denominator?
    Students may have already discovered that twelfths fit the gap between 2 and the answer on the number line.
    How many twelfths equal 3/4? How many twelfths equal 1/3? (Align fraction strip pieces if needed to check that 3/4 = 9/12 and 1/3 = 4/12)
     
  5. Use equations to model the problem:
    4 3/4 - 1 1/3 = 4 9/12 – 2 4/12
                         = 2 5/12 (subtracting ones first then twelfths)
    Why did we need a common denominator of twelve to solve this problem? (Addition and subtraction are only possible if the units are the same.)
     
  6. Write on the class/group chart: “What is meant by an equivalent fraction?”
    It is vital that students recognise that equivalent means of equal value, that is, different expressions of the same quantity. ½ and 5/10 are equivalent because they represent the same amount and occupy the same location on the number line between zero and one.
     
  7. Record student responses, including examples they give, e.g. 1/2 = 2/4
    Refer to the class/group chart recording from Session 1:
    4/4 = 1, 8/4 = 2, 12/4 = 3, 16/4 = 4, and the highlighted = sign.
    Do students agree that equals means “is the same as” or “that the amount on one side is equal in value to the amount on the other side”?
     
  8. Provide students with paper strips of the same length cut from photocopy paper.
    Use lengthwise folding to illustrate how equivalent fractions can be found.
    Here is the example of 2/3 = 4/6 = 8/12 = …

     
  9. Record equivalence equations, e.g. 2/3 = 4/6 = 8/12.
    What comes next in the equation? How do you know? (= 16/24 = 32/48 …)
    What patterns do you notice in the numbers?
    Students should notice that both the numerator and denominator are doubling.
    Why did that happen? (With each half fold the number of pieces that make one (denominator) doubled so the number of shaded pieces doubled)
  10. Provide two other examples of folding such as:
    Start with 3/4 and repeatedly halve to get 3/4 = 6/8 = 12/16
    Start with 1/2 and repeatedly third to get 1/2 = 3/6 = 9/18
     
  11. Provide your students with Fraction Strips (Copymaster One) and Pizza Pieces (Copymaster Two). 
    Find as many equivalent fractions as you can using these materials. Record the fractions using equations and diagrams. Create a poster of your work explaining how equivalent fractions work.

    Give students plenty of time to explore. Roam the room looking for:
    Are students confident with the meaning of numerator as a count, and denominator as the size of pieces counted?
    Do students anticipate the relationships among different sized pieces?
    Do students anticipate the effect of halving and thirding pieces of a parent fraction?
    Do students record equivalence appropriately using equations?
  12. After a suitable time gather the class to share posters. Look for patterns in the equations, particularly the multiplying and dividing of numerators and denominators by the same factor.
    For example: 
    • These fractions are equivalent
    • They are the same length
    • The fractions have the same value

Session Three

SLOs:

  • Recognise that equivalent fractions occupy the same place on the number line.
  • Recognise and apply the multiplicative relationship between simple equivalent fractions.
  1. Have students relook at their equivalent fraction posters. Together record on the class/group chart what the students notice about equivalent fractions highlighting that equivalent fractions:
    • have the same value
    • have the same length
    • have different numerators and denominators.
       
  2. Given each pair of students two strips of paper cut longways from A3 paper.
    Here are some equivalent fractions that I found:
    2/3 = 4/6 = 8/12
    I want to mark each fraction on a number line.
     
  3. Start by using a one length from the fraction strip set to mark zero and one respectively at the left and right ends.
    How will I find the place to mark two thirds? (Two copies of one third from zero)
    How will I find the place to mark four sixths? (Four copies of one sixth from zero)
    How will I find the place to mark eight twelfths? (Eight copies of one twelfth from zero)

    What other fractions will be equivalent to two thirds?
     
  4. Ask your students to create a number line showing some of the equivalent fractions that they identified. After a suitable time gather the strips and arrange them vertically on a wall or board.
    Look for vertical alignment to check for correct placement of fractions and equivalence.
     
  5. We have used lengths. I wonder if equivalence also works with areas.
    Provide each student with three blank A4 pieces of paper and ask them fold the pieces in these ways.
    You will need to think how the paper was folded.
    Remember to label each piece with the correct fraction.

     
  1. Gather the class.
    Use your paper pieces to show a partner that:
    • Three quarters are equivalent to six eighths (3/4 = 6/8).
    • Two thirds are equivalent to eight twelfths (2/3 = 8/12).
    • Five sixths are equivalent to twenty twenty-fourths (5/6 = 20/24).
       
  2. Allow student sufficient time to rehearse their demonstrations of equivalence. Look for signs of students noticing relationships among unit fractions, such as twenty-fourths are quarters of sixths.
     
  3. Share the results as a class, sticking the appropriate paper pieces on a board to check equivalence of area. You might try less obvious examples where the multiplying numerator and denominator by a whole number factor does not work. Good counter examples are:
    How many sixths are equivalent to three quarters? (3/4 = 9/12 and 1/6 = 2/12. There are 4 ½ sixths in three quarters)
    How many eighths are equivalent to one third? (1/3 = 8/24 and 1/8 = 3/24. There are 2 2/3 eighths in one third)
     
  4. Add to the points from the start of the activity:
    Equivalent fractions:
    • have the same value
    • have the same length or area
    • have different numerators and denominators
    • occupy the same place on a number line.
       
  5. Write on the class/group chart: 1/2 = 2/4 = 4/8
    Ask the students what they notice happening to the numerator (it is doubling each time) and what is happening to the denominator (it is doubling each time).
    Ask students how this could be shown. Accept their suggestions, reaching an understanding that it could be shown in this way.


     
  6. Together recognise that the value of 2/2  equals 1 and that when any whole number is multiplied by 1 the product equals that number. We can multiply a fraction by 2/2, 3/3, 4/4 etc. to find equivalent fractions without changing the value of the amount represented.
     
  7. Ask the students to play Fractions Fish, that can be made from Copymaster Three.
    Purpose: to recognise the multiplicative relationship between equivalent fractions)
    This game is for two or three players and the rules are included with the Copymaster.

Session Four

SLOs:

  • Understand that fractions can have an infinite number of names.
  • Understand the inverse nature of multiplication and division and how this knowledge can be used to simplify fractions.
  • Use multiplication and division to calculate equivalent fractions.
  1. Begin this session by reading together the final statement made in Session 3.
    Refer to this expression and have the students suggest how the pattern of equivalent fractions continues.

    How long can this pattern of making fractions that are equivalent to one half continue?

     
  2. Have students discuss this in pairs and agree on their answer. “Forever. We can just keep on going”.
     
  3. Ask: Which of these fractions would be in the list equivalent to one half?
    How do you know? (note that some fractions equal 1/2  but do not arise in the doubled list)
    50/100                         64/128             36/72               256/512                       22/44
    The common property of all fractions equivalent to 1/2 is that the denominator is twice the numerator.
     
  4. Discuss and conclude that there are an infinite number of names for any fraction. You might try other fractions such as 2/3 and 3/5 to see if you run out of possible equivalent fractions.  For example: 2/3 x 3/3 = 6/9 = 18/27 = 54/81 = …
     
  5. Write 9/12.
    Is there a simpler way to write this fraction?
    Let students investigate the question in pairs and record their ideas.
    Ideas might be:
    • We went ‘backwards’ (divided the numerator and denominator by 2 several times).
    • We know that 3/12 is the same as ¼.
    • We know that 9 is three quarters of 12.
       
  6. Create a fraction strip picture of the problem:

    If we look at the numerators and denominators of the fractions can we know the simplest fraction without using strips?
    Record the equality as:
    9/12 = 3/4
    We know that we can change three quarters into nine twelfths through multiplying by one. We use 3/3 as a name for one.
    What would undo multiplying both the numerator and denominator by three?
    Students might recognise that 9 ÷ 3 = 3 and 12 ÷ 3 = 4 which gives the numerator and denominator of 3/4.
     
  7. Provide some other examples of simplifying fractions by identifying a common factor. Use fraction strips to model each problem if needed. Note that students will need to consider twentieths as half of tenths, and twenty fourths as halves of twelfths.
    For example: 
    8/12 (2/3)                     6/10 (3/5)         15/20 (3/4)                  9/24 (3/8)    
        
  8. Summarise the findings about how to simplify fractions:     
    • To simplify a fraction, you can divide both the numerator and the denominator by the same whole number.
    • Multiplication and division are inverse operations, one undoes the other.
       
  9. Ask students to play Simplify Secrets in pairs.
    Purpose: To simplify fractions and to identify which fractions can be simplified.
    Student pairs share a deck of shuffled playing cards with picture cards and the jokers removed. They each need a pencil and a sheet of paper.
    Players take turns to turn over two cards. The first card is the numerator and the second is the denominator. The player records their fraction. Beside the fraction they write a simpler equivalent fraction if they can, showing what they divided both numerator and denominator by. If their partner agrees they tick their recording.
    Fractions for which they can find no simpler fraction are recorded and nothing further is added.
    For example:
    If a player draws a 6 and then an 8, they write 6/8 = ¾, justifies the division factor, and once their partner agrees they can add a tick
    If a player draws a 7 and then a 9, they write 7/9 but no tick is given. 
    If the fraction is an improper fraction that is changed to a mixed numeral without simplification, no tick is given as this is not simplifying the fraction. For example: 7/2 = 3 ½ earn no tick bit 9/3 = 3/1 does earn a tick. 
    Students play and record 10 rounds (at least). They should then look closely at the fractions they were able to simplify () and the ones they were not.
    They must discuss, agree, and write the ‘Secret to simplifying fractions is…’
     
  10. Bring the class together after the game to generalise that a fraction can be simplified if both numerator and denominator share a common factor.
     
  11. For practice in finding equivalent fractions ask students to independently complete MM8-9 part 1, Equivalent fractions: Equation and Explanation. 
     
  12. On the class/group chart summarize what has been learned in this session.
    For example:
    • A fraction can have an infinite number of names if we keep on multiplying the numerator and denominator by the same number.
    • Some fractions can be simplified by evenly dividing the numerator and denominator by the same number.

Session Five

SLOs:

  • Find the common multiple and use this to create equivalent fractions.
  • Apply knowledge of equivalent fractions to solving problems which involve comparing, adding and subtracting fractions with different denominators.
  1. Begin by posing the question:
    When would it be useful to know about equivalent fractions?
    Record the students' suggestions and ask them to provide examples. They may offer problems like these:
    • If we are comparing different kinds of fractions to see which fraction is greater. Like you might have 3/4 of metre of fabric and you need 2/3 metre. Will you have enough?
    • If we are adding different fractions. Say you had 3/4 of a pizza and another 1/3 of a pizza. Does that make 1 whole pizza?
    • If you have an amount and gave away a fraction, and you want to work out what you have left. Like you have 5/8 of something and you gave away 1/4. 
       
  2. Provide these two problems for students to solve in pairs, before sharing with the class.
    • You have two jobs to do. Mowing the lawn takes 3/4 of an hour. Washing and polishing the car takes 2/3 of an hour.
      How much time, in hours, does it take to complete both jobs?
      (3/4 + 2/3 = 9/12 + 8/12 = 17/12 = 1 5/12)
    • The bottle has 1 4/5 litres of juice left in it. You pour 3/4 of a litre from the bottle.
      How much juice is left in the bottle?
      (1 4/5 - 3/4 = 9/5 - 3/4 = 1 16/20 – 15/20 = 1 1/20)
       
  3. Model each problem with fraction strips. Discuss:
    Why is finding a common denominator necessary to solve both problems?
    Which common denominators did you use? Why did you choose those numbers?
    When did you use equivalent fractions to solve the problems?
     
  4. Pose the question:
    What is the best way to find a common denominator?
    Let’s use 2/3 + 4/5 as an example.
    From previous work students might know that thirds and fifths can be equally partitioned into fifteenths.
    How many equal parts is each third broken into, so that fifteenths are formed? (five)
    You may need to demonstrate the partitioning with fraction strips. Students need to see that the size of the parts comes from considering how many of those parts make one (whole).
    How many fifteenths are equivalent to two thirds? (write 2/3 = 10/15)
     
  5. Ask the same questions for four fifths (Fifths are cut into three equal parts to form fifteenths so 4/5 = 12/15)
     
  6. Summarise:
    We look for the least common multiple of 3 and 5 (LCM) so both fractions can be named with the same denominator.
     
  7. Provide students with Copymaster Four which is a sheet of hundreds boards. Show how the common multiples of 3 and 5 can be found. Students can copy you on their page.
    Shade the multiples of three in one colour; 3, 6, 9, …
    Shade the multiples of five in another colour; 5, 10, 15,…
    Continue until all the multiples have been found.
    How do we know which numbers are common multiples of 3 and 5? (double colour)
    What is the least common multiple of 3 and 5? (15)
    Let students work on these tasks using Copymaster Four as support.
    • Find the common multiples of 2 and 3?
    • Find the common multiples of 2 and 5?
    • Find the common multiples of 5 and 6?
    • Find the common multiples of 4 and 6?
    • Find the common multiples of 7 and 13?
       
  8. Discuss the results. Pay attention to:
    Why are the common multiples of 2 and 5 in the right-hand column?
    Why is 12 the LCM of 4 and 6, when 4 x 6 = 24?
    Why do 7 and 13 have only one common multiple less than 100?
     
  9. When we solve a problem like 1/3 + ½ = what common denominator might we use? Why?
    Note that any common multiple will work but LCM is the simplest to use.
    Record the thinking using a table like this:

     
  10. Use the same format to capture finding the common denominator for the problems in section 2.
    3/4 + 2/3 =
    1 4/5 - 3/4
     
  11. Ask your students to do more work with Common Multiples (Copymaster Five). Some students will need support to understand what is required. If necessary, model solving the first problem then encourage students to work independently.
     
  12. Roam the room and look for:
    Do your students look for pattern in the occurrence of multiples?
    For example, the common multiples of 3 and 8 occur in every eighth cell in the 3 column and row, and every third cell in the 8 column and row.
    Can they identify the LCM in each problem?
     
  13. Have students play the game, I can make it, in pairs. (Copymaster Six).
    Purpose: To apply knowledge of equivalent fractions to solving problems.
    This is a game for two players. Story cards and fraction cards are shuffled and placed face down in two piles. Each player takes two story cards and 6 fraction cards and places them face up in front of them.
    Players take turns to:
    • Check if one of their fraction cards allows them to answer one of the problems, one the problem card. If so, the player matches the two cards and puts the two cards together as a pair. Player Two checks the match.
    • Whether they can make a pair or not the player adds an extra fraction and problem card from the two piles, or they propose a swap of one of each type of card with the other player.
      The winner is the player that makes the most pairs.
      Note: There will be 7 fraction cards left over because some are ‘trick’ cards.
       
  14. Ask each student to write at least three contextual addition or subtraction of fractions problems for a partner to solve. Each problem must have a model answer. Make a class book of addition and subtraction problems.

Extra work:

Students could work on some of the Fractions e-ako to consolidate learning of concepts developed in these lessons. These e-ako are at the right hand end of the level 3 shelf on the Additive thinking and Multiplicative thinking pathways.

Students work independently to complete MM8-9 - part 2, explanation of which fraction is larger. Ask them to pair, share, and discuss. 

Students watch the following videos to see how a unifix cubes model can be used to add and subtract fractions:

Printed from https://nzmaths.co.nz/user/75803/planning-space/late-level-4-plan-term-1 at 1:12pm on the 21st January 2022