Late level 4 plan (term 3)

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
Number and Algebra
Units of Work
This unit helps students to develop procedural fluency with integers and have conceptual understanding of integers in the real world.
  • Understand everyday application of integers.
  • Add and subtract positive and negative integers.
  • Use models to explain why subtraction of a negative integer has a positive effect.
Resource logo
Level Four
Number and Algebra
Units of Work
This unit supports students to recognise percentages as equivalent fractions, and to carry out simple calculations involving finding percentages of amounts.
  • Use the percentage bar model to find the percentage that a part is of a whole.
  • Use the percentage bar model to find a percentage amount of a whole.
  • Simplify parts of a whole to common fractions to find percentages.
  • Use percentages to represent the relationship between two different wholes...
Resource logo
Level Four
Geometry and Measurement
Units of Work
In this unit students make and investigate various solids, including regular and semi-regular polyhedra, and cylinders and cones. They look for patterns in the numbers of faces, edges and vertices they see whether they can “discover” Euler’s famous formula. By truncating the vertices of the Platonic...
  • Construct models of polyhedra using construction materials, like geoshapes or polydrons.
  • Use the terms faces, edges and vertices to describe models of polyhedral and look for relationships between these features.
  • Anticipate the features of the solid created when a Platonic solid is truncated...
Resource logo
Level Four
Number and Algebra
Units of Work
This unit requires students to apply their number sense about the size of decimals to estimate and calculate the product of decimal fractions. In doing so they generalise about the effect of multiplying and dividing by ten and one hundred.
  • Express a multiplication of two decimals as a product of fractions, e.g. 0.4 x 0.7 as 4/10 x 7/10 = 28/100.
  • Connect the product of the two fractions to the decimal answer, e.g. 28/100 = 0.28.
  • Know the effect of multiplying and dividing a decimal number by ten or one hundred.
  • Use...
Resource logo
Level Four
Integrated
This unit provides a set of learning tasks that integrate across the strands of the Mathematics and Statistics learning area of the New Zealand Curriculum and provide opportunities for assessment of student achievement across those strands. Each session may involve more than one lesson, especially...

Session One
Use systematic approaches to find all the possible outcomes, e.g. tree diagrams, organised lists.

Session Two
Use tables, graphs, and word rules to represent growing patterns.

Session Three
Draw cube models using plan views.

Session Four
Draw cube models using isometric...

Source URL: https://nzmaths.co.nz/user/75803/planning-space/late-level-4-plan-term-3

Integers

Purpose

This unit helps students to develop procedural fluency with integers and have conceptual understanding of integers in the real world.

Achievement Objectives
NA4-2: Understand addition and subtraction of fractions, decimals, and integers.
NA4-6: Know the relative size and place value structure of positive and negative integers and decimals to three places.
Specific Learning Outcomes
  • Understand everyday application of integers.
  • Add and subtract positive and negative integers.
  • Use models to explain why subtraction of a negative integer has a positive effect.
Description of Mathematics

Integers are needed to meet the demands of situations where a larger whole number is subtracted from a smaller whole number. For example, if a person has $5 available in cash but owes $8 then their net situation is 5 – 8 = -3. The origins of integers historically lie in the algebra of subtraction of whole numbers. Integers, as quantities, usually reflect a state of balance between directional forces, such as cash being and asset and debt a liability. Furthermore, integers are sometime called directional numbers because they represent a magnitude (size) and a direction, e.g. -3 represents a magnitude of 3 units in a negative direction from zero. The directional nature of integers is important to real world applications such as transmission factors of gears and pulleys, and to enlargement (dilation).

A common use of integers in real life is to label and quantify points on a scale, such as temperature and height above sea level. In both cases the location of zero is important to the attribute being measured. For example, both the height above normal sea level (0 m) of a spring tide and the temperature below the freezing point of water (0°C) have significant consequences to the severity of the situation. Zero acts as an important benchmark indicating normality or balance. This is also true in sport or games like Bridge where negative numbers reflect a state relative to expectation, e.g. -6 in golf means six under par when a player has taken six fewer shots than the expected norm.

Specific Teaching Points

Integers are an extension of the whole number system. Therefore, the properties of Integers under the four operations should be the same as those for whole numbers. With addition and subtraction four main properties hold:

The commutative property of addition

The order of the addends does not affect the sum. If -3 + 4 = 1 then 4 + -3 = 1. Note that the commutative property does not hold for subtraction. For example, 4 - -3 = 7 but -3 – 4 = -7.

The distributive property of addition

This property is really about the partitioning of addends and recombining those addends. For example, if 5 = -1 + 6, then -2 + 5 = (-2 + -1) + 6. This property does not hold for subtraction.

The associative property of addition

This property is about ‘associating’ pairs of addends one pair at a time. For example, (-4 + 3) + -1 = -4 + (3+ -1). This property does not hold for subtraction.

Inverse operations

Addition and subtraction are inverse operations so one operation undoes the other. For example, -2 + -3 = -5 so -5 - -3 = -2.

It is the need for these number laws to hold that establishes the effect of operations, such as subtracting a negative integer has the same effect as adding a positive integer.

This unit combines two of Hans Freudenthal’s (1983) models for operations on integers, the annihilation and vector models. In the annihilation model positive and negatives cancel each other, so +1 and -1 pairs equal zero. The act of creating or removing one positive and one negative pairs that equal zero does not alter the quantity being represented. The vector model presents integers as magnitudes with direction. +1 is represented by a vector of length one in a positive direction and -1 as a vector of length one in a negative direction. Freudenthal cautioned that a quantity of +1 or -1 was easily confused with the operation of adding or subtracting one and teaching needed to make that difference explicit.

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:

  1. Physically moving up and down a number line to act out addition and subtraction as operations. Addition involves facing right and subtraction involves facing left. The sign of the integer tells whether to walk forward (positive) or backwards (negative).
  2. Using the physical and diagrammatic models in conjunction with symbols so students can work out the answers to calculation by linking the symbols to quantities, e.g. cash as positives and debts as negatives.
  3. Supporting students by modelling correct equations for calculations. Be clear about the difference between an operation symbol, and a direction symbol such as -4 which gives a direction of movement and the size of that movement.
  4. Use calculators in a predictive way, that is thinking about the answer using models, then testing the answer out on a calculator. Also use calculators to generate patterns of equations quickly, and expect students to generalise from the patterns. For example, subtracting a negative as the same effect as adding the matching positive.

The unit uses the contexts of money, positive and negative spaces, and scale. Other contexts might better suit the interests and cultural backgrounds of your students. Students interested in sports might enjoy golf as a context, and those who enjoy computer find points schemes interesting. Some students may enjoy the context of comparing temperatures from locations around the world, and finding out which locations have the most and least variation in a full year.

Activity

Session One

This session introduces negative integers in real life and presents integers as vectors.

  1. Begin with PowerPoint One that shows some situations where negative numbers are used. The contexts include temperature above and below zero degrees, water level above and below sea level, time before and after the birth of Christ, and financial well-being as a balance of available money and debt.
  2. For the next activity you will need sets of cards made from Copymaster 1. The cards will be used throughout the unit so investing time in laminating sets is worthwhile. Go outside and draw a horizontal number line on the pavement like this.
  3. Ask students how the number line might be extended. The real-life contexts should encourage them to consider the placement of zero and the negative integers. In the end you want a number line from -8 to 8. Mention that usually we do not write +8 for 8 though we could.
    Ask one student to act out the addition of integers as vectors. Addition is the operation of combining quantities. Ask him or her to begin on zero and act out the cards they are given. Draw a set of three cards randomly (made from Copymaster 1). The +1 card represents one unit in a positive (right) direction and -1 represents one unit in a negative (left) direction. As the student walks out the movement given by the cards explain to the whole class why they are moving in that direction and the distance of that move.
    Act out several combinations of cards then ask the students if it is possible to predict the finishing location if the cards are known. Choose a set of five cards and invite predictions.
    What do you notice about what happens when +1 and -1 cards combine?
    Is that also true of -1 and +1?
    How can the fact that positive and negative cards form zeros be used to predict the finishing location?
  4. Return inside for students to play a game of “Which way Wally Weta?” in pairs or threes. Photocopy the gameboard (Copymaster 2). Teams will need counters, preferably transparent, and a set of cards with equal numbers of +1 and -1. Each turn the cards are shuffled and the top four cards turned over one at a time. With a counter the player acts out the moves, starting at zero. They finish each play by leaving the counter at the finishing number on the board. Play continues like that for 16 turns.
    The ‘house’ gets any counters on -2 or +2 and the team gets any counters that are left.
    Is the game fair?
  5. Invite students to discuss whether the game is fair, that is their team and the house have an equal chance of winning. You might look at the results of some games to see what seems to happen.
  6. Work ‘outside in’ to work out the ways of the counter finishing on a particular number.
    What cards will give +4? What about -4? (You might record -1 + -1 +-1 + -1 = -4 or 4 x -1 = -4.)
    What cards will give you +3 or -3? (Impossible? Why?)
    What cards will give you +2? (+1, +1, +1, -1) How many different orders can the cards come in?
    How can you use the number of ways to get +2 to find the ways to get -2?
    What other finishing numbers are possible? (Only zero – Why?)
    How many orders of cards will give zero?
  7. The game is fair as zero can be arrived at in six different card orders, meaning that the house has eight out of sixteen outcomes for them and the player has the other eight outcomes.
  8. A nice way to show the possibilities is to consider the ‘ordinal positions’ of the → cards only. The ← cards must take up the in between positions. Order of the cards being collected is important in determining the different ways a counter can finish at a particular place. You may need to drop back to a simpler form of the game, e.g. two or three cards to support the students with ways to find all the outcomes.
    This table shows all the possible outcomes for the four card game:

Card One

Card Two

Card Three

Card Four

Total

4

2

2

2

2

0

0

0

0

0

0

-2

-2

-2

-2

-4


Session Two

In this session students explore the ‘Hills and Dales’ context for application integers. The context was used in the Oscar nominated film “Stand and Deliver” about Jaime Escalante, an American teacher working with disadvantaged students in Los Angeles. A short video of him teaching algebra using the ‘Hills and Dales’ model is easily accessed online. The video finishes with Escalante asking his students why a negative number multiplied by a negative number gives a positive answer. Good question!

In this unit the context is about road builders. In real life one of the largest costs of new roads is relocation of earth, particularly if earth must be brought in from off-site. Roads are best flat and both hills and dales present potential costs unless a hill can be used to fill a dale.

  1. Work through PowerPoint Two A shows some scenarios in which a hill (+1) might fill a dale (-1) to create flat land (0). With each slide ask the students two questions:
    What will happen in this situation?
    What is the equation for this situation?
    For example, scenario one is three hills (+3) and two dales (-2). You might connect that scenario to the arrow cards by getting three +1 cards and two -1 cards. Students should recognise that you are combining integers and offer the equation 3 + -2 = 1 or -2 + 3 = 1. In fact, noting the commutative property holds with Integers is important. Work through the four scenarios before giving the students Copymaster 3 to complete independently.
  2. For early finishers pose this problem:
    The foreman notices that there are three more dales than hills.
    Draw some landscapes where that would be true.
    What is the same about all the landscapes you could draw?
  3. Process the students’ answers to Copymaster 3 with emphasis on how one positive and one negative cancel out each other to form a zero. Ask what other situations in real life are like that. Situations might include having money but owing money, scoring below par and above par in golf, getting points and penalties in video games, putting hot air in the balloon and adding weights, going up in an elevator and going down, eating a hamburger and exercising hard.
  4. To further practice addition of Integers ask the students to play games of Integer Coverup
  5. More practice with the Hills and Dales model is available in the Using integer tiles activity

Session Three

In this session students explore the addition of Integers in the context of dollars and debts. The net financial position of a person is the sum of the money they have available and the debts they owe. 

  1. Pose this scenario to the students:
    Layla has $7 in her bank account. She owes her parents $3. How well off is Layla?
  2. Students should tell you that Layla is worth $4 which would be the result of paying her parents off and being debt free. Her net position can be shown in many ways:
    • Using toy money and IOUs

       
    • Use vectors on a number line with +1 representing $1 and -1 representing a bill for $1.
  3. Whatever model is used the key point is annihilation of a one dollar debt by a one dollar coin and vice versa. Pose similar problems staying with the Layla scenario and changing the amounts. Be sure to include dollars and debts that result in a negative net position, e.g. $10 money and $13 in debts.
  4. Ask the students to practice this idea using the Disappearing dollars Figure It Out activity.
  5. Extend the dollars and debts model to subtraction of Integers. Begin with Layla’s scenario with a subtle change.
    Layla has $7 in her bank account. She owes her parents $3. Her parents say they will not need her to repay $2 of the debt. How well off is she now?
    This story can be represented by the equation 4 - -2 = 6. Originally Layla’s position was $4, accounting for her money and debt. Removal of a two dollar debt is represented by - -2.
  6. Ask: Is Layla better or worse off by her parents’ decision?
  7. Model the scenario using vectors as below.
    • Initial situation
    • After the removal of $2 of debt.
  8. Pose similar problems to get a balance of starting position (positive or negative) and subtraction of both debts and dollars (spending). Adapt the Layla scenario. For example:
    Layla has $8 in her bank account. She owes her parents $13. Her parents say they will not need her to repay $5 of the debt. How well off is she now? (-5 – -5 = 0)
    • Initial state -5
    • Final state 0
  9. To practise subtraction using the money model students can work individually or in pairs through the e-ako called "AS4.60 More Integers (Positive and negative numbers)".

Session Four

In this session the vector model is connected to the Hills and Dales and Dollars and Debts models. The aim is to generalise addition and subtraction of Integers. It is important to distinguish the vectors that represent positive and negative numbers and the addition and subtraction as operations, addition as movement to the right and subtraction as movement to the left.

  1. Begin with slide one of PowerPoint Four A. Ask: What do these models have in common?
    Invite students to give the balance of -2. Ask: Where is -2 is in each model?

    If we start with a balance of -2 and subtract 3, what does that look like in each model?
  2. Slides 2-4 of PowerPoint Four A are animated to show what subtracting one looks like on each model. Record the operation as: -2 – 3 = -5
  3. Ask the students to anticipate the results on the three models of removing different amounts, particularly 2, 1 and 0. Arrange the equations in order.
    -2 – 3 = -5
    -2 – 2 = -4
    -2 – 1 = -3
    -2 – 0 = --2
    What happens if -1 is subtracted (removed)?
  4. Slides 5-7 of PowerPoint Four A model the operation that leaves the balance at -1. Continue the pattern to -2 – -3 = -1. Discus the positive effect of removing a negative amount, that is the balance is greater than before (There is more earth and more money).
  5. Provide the students with a set of similar equations but starting with a positive balance, such as:
    +3 – 2 = -_
    +3 – 1 = -_
    +3 – 0 = -_
    +3 – -1 =  _
    +3 – -2 =  _
    Ask them to justify their answers using one of the models that have been used.

Session Five

In this session the vector model is developed into a number line model which highlights the direction of change when Integers are added and subtracted.

  1. Start with these True/False statements. Ask the students to discuss both statements in small groups.
    When two numbers are added the sum is always greater than the number you start with. True or False or Sometimes True
    When a number is subtracted from another the difference (answer) is always less than the starting number. True or False or Sometimes True
  2. After a suitable period of discussion share ideas. Look for students to state the conditions under which the statements are true or false. The first statement is true when the second addend is greater than zero (a + b > a iff b>0) and is false if the second addend is equal or less than zero (a + b ≤ a iff b≤0). Find similar conditions for the second statement, i.e. a - b < a iff b>0 and a - b ≥ a iff b≤0. Note that iff means "if and only if">
  3. Ask the students to work in pairs through the e-ako called "AS4.50: Integers (Positive and negative numbers)". They will learn about the direction of change through the context of Claw the Crab.
  4. Summarise the addition and subtraction of integers using this diagram.
  5. To practise the direction of change play the game Walk the Plank with students . You may like to amend the game so the dice are labelled + (for the direction of the shark, S) and – (for the direction of the boat, B). The activity could be used as a probability task where the fairness of the game is examined both experimentally and theoretically. The usual length of a game in number of rolls before the pirate is safe or wet could also be investigated.
Attachments

Getting partial to percentages

Purpose

This unit supports students to recognise percentages as equivalent fractions, and to carry out simple calculations involving finding percentages of amounts.

Achievement Objectives
NA4-3: Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.
NA4-5: Know the equivalent decimal and percentage forms for everyday fractions.
Specific Learning Outcomes
  • Use the percentage bar model to find the percentage that a part is of a whole.
  • Use the percentage bar model to find a percentage amount of a whole.
  • Simplify parts of a whole to common fractions to find percentages.
  • Use percentages to represent the relationship between two different wholes or parts.
Description of Mathematics

In this unit we build on work by Prediger and Pohler (2015) to develop students’ concept of percentages using the metaphor of a data download. The model is linear which has been effective in the teaching of fractions and decimals.

Percentages are a specific type of equivalent fraction with a denominator of 100. The symbol %, is derived from the vinculum of a fraction /, and the two zeros from 100. So, a percentage literally means “for every hundred.”

In many situations a percentage describes a part-whole relationship. For example, Mani lands 24 out of 30 shots in her netball game. What is her shooting percentage? The calculation of her percentage is 24/30 = ?/100. There is an assumption of homogeneity, that Mani will shoot at the same rate.

Percentages are very useful to establish a common base for comparison. Most situations in which part-whole relationships need to be compared, the whole (or bases) are not the same. For example, Mani takes 30 shots, but Alicia takes 28 shots. Some situations involve comparison of different wholes or different parts. In those situations, percentages greater than 100% are possible. For example, there are 12 girls and 18 boys. The number of boys is 150% of the number of girls.

In other situations, percentages are used as a fractional operator, e.g. 30% of $45. That calculation is equivalent to 30/100 x 45 = ?. Common examples of percentages as operators are discounts in shops, and interest on loans.

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. The difficulty of tasks can be varied in many ways including:

  • Alter the numbers that you choose for the problem. Easier problems involve simple percentages such as 50%, and 10%, moving to derivatives of those percentages like 25%, 5%, and 30%. Use base amounts that make the calculations simpler, such as 60  or 30, before using more complicated bases such as 24, 36 and 144.
  • Ensure that student understand the meaning of important vocabulary, such as percentage as “in every hundred”, base as the unit of comparison, and amount as the target quantity.
  • Diagrammatically model the problems using the percentage bar. Explicitly demonstrate use of the model with students creating their own diagrams as you work. The percentage strip is a powerful way to scaffold students’ representation of percentage problems.
  • Use calculators in a predictive way. Expect students to estimate or calculate their answer initially, using percentage bar models, before confirming the answer using a calculator. Students should be shown how to perform algorithms on a calculator to find percentages, e.g. 45% of 64 as 64 x 45% =

The contexts used in the unit can also be adapted to cater for the cultural backgrounds and interests of students. Choose situations that are likely to be familiar to your students. The unit uses situations around shopping, download bars, and popular travel destinations, which will appeal to many students. Other situations such as sharing collected food, a walk or cycle between two local points, battery charge remaining in a computer game, or sports points, might be more motivating to your students.

Required Resource Materials
  • Dot paper, or geoboards and rubber bands
  • Scissors and glue sticks
  • Copymasters One, Two, Three, Four, Five, and Six
  • PowerPoints OneTwo, Three, and Four
  • Access to computer spreadsheets
Activity

Session One

  1. Tell the students that they will be investigating percentages in this unit. Ask them for examples of when percentages are used in everyday life. Slide One of PowerPoint One can be used if needed to inspire more ideas. Common contexts are:
    • Sale in shops, e.g. 25% off
    • Sports, e.g. kicking/shooting at 80%
    • Computer downloads, e.g. 75% of 4mb downloaded
    • Livestock births, e.g. lambing percentage of 180%
    • Inflation or interest, e.g. mortgage at 5% interest
    • Statistics, e.g. 46% of the people choose hokey pokey icecream
  2. Ask: When we go to a 25% off sale, how much is taken off the normal price?  (one quarter)
    How much of the normal price do we pay? (75% or 3/4)
  3. Write these two equations on the whiteboard; 1/4 = 25/100 and 3/4 = 75/100. Discuss that the fractions are equivalent.
    What does it mean when you pay 75% of the normal price, but the item does not normally cost $100? Say it costs $60.00?
  4. Discuss that three quarters of the normal price is paid, whatever the normal price is. Take the 75% of 60 example and write; 75/100 = 3/4 = ?/60.
    What number replaces the question mark to show how much you pay for the item?
    Students should use their knowledge of equivalent fractions, and recognise that multiplying both the numerator and denominator of three quarters by 15 gives 45/60. So, the item costs $45 at the sale.
  5. Look at Slide Two of PowerPoint One which shows a file download status bar with percentages. Discuss what the bar shows and that the percentages represent the fraction of the data file that is downloaded. For the next three slides discuss what the percentage message is likely to be. Look for students to identify known fractions and try to represent them as percentages. For example, Slide Three shows 10%. Students might realise that five ‘iterations’ of the yellow bar make one half. Therefore, the bar shows one tenth. For each bar write equivalent fraction equations.
  6. Give the students copies of Copymaster One to work on in pairs or individually. Look for your students to:
    • Use benchmarks like 50%, 25%, and 75% to estimate the percentages;
    • Identify the fractions of the whole bar that are shaded, and use equivalence to find the percentages;
    • Iterate a trusted unit like 10% (one tenth) to approximate the percentage;
    • Possibly use a ruler to measure the length of the whole bar and shaded section to approximate the percentage.
  7. After an appropriate time, gather the students together to share their strategies and answers. Highlight the points above in the discussion. Finally, provide one copy of Copymaster Two to each team of students. Challenge them to match the fraction, decimal, and percentage cards. For example, 0.2, and 20% go together. Ask each team to make a chart by pasting each grouping together and drawing a percentage download bar to match.
    Students must be able to justify the percentage to fraction equivalence. For example:
    • T: How do you know that one quarter equals 25%?
    • S1: 25 multiplied by four equals 100 so 25% is one quarter of 100%
    • S2: 50% equals one half, so 25% equals one quarter since half of 50 is 25.

Session Two

In the next two sessions, students explore using the percentage bar as a model for solving a variety of part-whole percentage problems. The language of rate, amount, and base are introduced. Calculation steps are derived from the strategies that students use when solving problems with the percentage bar.

  1. Begin with the download animation to reinforce the fraction to percentage connections from Session One. Run the animation on Slide One of PowerPoint Two with instructions:
    Put your hand up when (fraction) of the data is downloaded.
    Justify why most of you chose this spot (pointing to download bar).
    Vary the fractions to include quarters, thirds, fifths and tenths.
  2. Slides Two through Seven of the PowerPoint Two introduce problems of finding a percentage of an amount, using the percentage bar as a visual model. For each problem let the students attempt to solve the problem and discuss the various ways it might be solved. For example, 80% of 35 could be solved by finding 10% first, then multiplying that amount by eight. It might also be solved by finding 20% first then subtracting that amount from 100%.
  3. Every second slide shows how the percentage bar can be used to track and organise calculations. The slides also show the calculation in equation form. You may need to remind students of the process of finding a fraction of a whole number, using simple examples first. It is important to be clear about the meaning of the terms, base, rate, and amount. For every example, bring students back to the meaning of the words in context.

    Base means the unit of measurement or comparison. The percentage is calculated as a fraction of the base. Sometimes the fraction is more than one, so the percentage is greater than 100%.

    Rate means a relationship between two measures. The relationship is multiplicative not additive. The word per is used to mean ‘for every’. For example, 60 km/h means 60 kilometres pass for every hour. 75% means 75, of the amount, for every 100 of the base.

    Amount means the result of applying the rate to the base. For example, if the base is 40 and the rate is 25% then the amount is 25% of 40 = 10.

  4. Slides Eight to Eleven present a different scenario, shopping. Invite the students to use a percentage bar diagram to solve each problem. The second slide for each problem shows possible solution strategies on a percentage bar and as an equation.
  5. Give students Copymaster Three to work from, individually or in pairs. The worksheet provides a variety of percentage problems about part whole relationships in different contexts. Look for your students to:
    • Use the percentage bar model effectively to represent the problem and work towards a solution.
    • Recognise what is missing in the problem, the base, the rate or the amount.
    • Write equations to represent the calculations that solve each problem.

Session Three

  1. Remind the students of the meaning of the words base, rate and amount. Show slide one of PowerPoint Three
    Here is a percentage problem for you to solve.
    What is the base? (24 people)
    What is the amount? (from 9, 5, 4, and 6)
    What is missing for you to find out? (the rate or percentage)
  2. Ask your students to work in pairs to work out the correct percentages. Use Slide Two to work through the answers. Students should recognise two main strategies:
    • Convert the part-whole fraction to a simple fraction that they know the percentage of, e.g. 6/24 = 1/4 = 25%.
    • Use trusted percentages, equally partition and combine them to reach the required amount, e.g. 50% of 24 = 12, 8 is two thirds of 12 so they need two thirds of 50%.
  3. Provide students with copies of the percentage wheel (Copymaster Four).
    Ask them to use the marks on the wheel to create a pie chart of the holiday destinations from Slide One.
  4. Let your students try to mark out sectors of the circle that match the percentages they have calculated.
    What fraction of the circle have you given to Fiji? Explain why you did that.
    Samoa was chosen by 25% of students. What fraction is that?
  5. Slide Six shows a model answer. Discuss strategies for accurately drawing the sectors. You might show how a spreadsheet can be used to draw the pie chart. The movie on Slide Seven shows how to do that and show percentages in each sector.
  6. Distribute the tally sheets from Copymaster Five around the room. Tell your students to visit each chart and add a tally to it, that shows their preference. You may need a short lesson on tally charts. Students must choose one of the options for each table.
  7. When the data is collected, allocate one chart to each group of three or four students. Their task is to calculate the percentage for each category. Encourage the use of the percentage strip as a model. Allow the use of calculators.
  8. Once students have calculated the percentages, they can draw their pie chart using the percentage wheel, then use a computer spreadsheet to create a pie chart with percentages. Look for students to:
    • Apply appropriate benchmarks to estimate the percentages.
    • Use the terms base, rate and amount correctly for their data.
    • Create percentage bar diagrams or equations to support their calculations.
    • Check that their calculations and pie charts match that generated by the spreadsheet.

Session Four

This session extends application of the percentage bar to more complex percentage problems. The previous sessions were about part-whole situations. This session introduces the use of percentages to compare two different whole or two different parts.

  1. Begin with this problem:
    In this story we have two files to download at the same time. One file is 12 kilobytes in size and the other is 18 kilobytes. Let’s look at what happens if the files download at the same rate, in bytes per second.
    Slide One of PowerPoint Four has an animation of the respective downloads. You may want to play it twice or three times.
    What did you notice about the percentages?
    Students might notice that the percentage for the smaller file is one and one half times the percentage for the larger file. Note that 1 ½ = 3/2. The percentage for the larger file is two thirds the percentage for the smaller file. Note that 2/3 is the reciprocal of 3/2, meaning the numerators and denominators are swapped.
    What will happen if the animation bars keep growing until the large file is 100% loaded?
    What will the percentage be for the smaller file?
  2. Play the animation on Slide Two then work through Slides Three and Four to show percentage bar models of the relationships. The two different selections for the base lead to different percentage relationships. With the equations ask:
    Is the percentage more or less than 100%? How do you know?
    ​​​​​​​Expect students to recognise that if the base is greater than the amount then the percentage is less than 100%. If the base is less than the amount, the percentage is greater than 100%.
  3. Slides Five through Eight provide other download scenarios in static form. For each example, ask your students to write the relationships using percentages. Encourage them to use percentage bar models if they need to. Allow the use of calculators. Look for your students to:
    • Use common factors to partition the bars into equal parts, e.g. both 12 and 30 divide into threes;
    • Express the relationships using simple fractions, e.g. 12 to 30 can be expressed as 4 to 10;
    • Use the fraction relationships to find the percentages, e.g. 4/10 = 40% and 10/4 = 2 ½ = 250%.
    • Use division on the calculator, to check and to calculate the answers, e.g. 12 ÷ 30 = 0.4 = 40% and 30 ÷ 12 = 2.5 = 250%.
  4. Copymaster Six provides many contextual examples of whole to whole or part to part comparisons. It begins with simple percentage strip diagrams and progresses to everyday situations. Check to see that your students apply the ideas in the bullet points above.

Solid Understanding

Purpose

In this unit students make and investigate various solids, including regular and semi-regular polyhedra, and cylinders and cones. They look for patterns in the numbers of faces, edges and vertices they see whether they can “discover” Euler’s famous formula. By truncating the vertices of the Platonic solids, they find a class of semi-regular polyhedra. They also generalise how to know if arrangements of polygons will create a closed polyhedron and the common properties of nets for prisms and pyramids.

Achievement Objectives
GM4-5: Identify classes of two- and three-dimensional shapes by their geometric properties.
GM4-6: Relate three-dimensional models to two-dimensional representations, and vice versa.
Specific Learning Outcomes
  • Construct models of polyhedra using construction materials, like geoshapes or polydrons.
  • Use the terms faces, edges and vertices to describe models of polyhedral and look for relationships between these features.
  • Anticipate the features of the solid created when a Platonic solid is truncated (its vertices cut off with straight cuts).
  • Anticipate if an arrangement of regular polygons around a vertex will create a bounded polyhedron.
  • Create nets for regular and semi-regular polyhedra using knowledge of the faces and symmetry.
Description of Mathematics

A polyhedron (singular) is a three-dimensional solid object which consists of a collection of polygons that bound a space. That means that the space is fully enclosed by the polygons. The simplest polyhedra (plural) are created by joining regular polygons, such as equilateral triangles, squares and regular pentagons. This family of polyhedra are known as the Platonic solids, named after the Greek mathematician Plato (though actually proved by Euclid).

There are 5 platonic solids, the cube (6 squares, 3 meeting at each vertex), the tetrahedron (4 triangles, 3 meeting at each vertex), the octahedron (8 triangles, 4 meeting at each vertex), the dodecahedron (12 pentagons, 3 meeting at each vertex), and the icosahedron (20 triangles, 5 meeting at each vertex).

Terms commonly used to describe the attributes of polyhedra include:

  • Face: A single polygon in a solid figure
  • Edge: A line where two faces connect
  • Vertex: A point of intersection of edges – a corner

In the 1750’s Leonhard Euler discovered a famous relationship between these three properties. The number of vertices, plus the number of faces take away two equals the number of edges.

E = V + F - 2

This unit involves exploration with three dimensional shapes and would be ideal as a lead in to the unit Building with Triangles, a level four unit which goes on to look at a group of polyhedra, the Platonic solids, in more detail.

This unit builds on the Level Three unit called Polyhedra (3D Shapes). It extends students’ knowledge of polyhedra to include Euler’s theorem and the properties of shapes around a vertex for semi-regular solids.

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 be supported at the station tasks (note that students who are more capable at constructing models are not necessarily also more capable at identifying patterns in attributes)
  • providing pre-made versions of models and nets that students can refer to when making their own
  • restricting the numbers of models and nets that students are asked to make.

This unit is focussed on the construction of specific geometric shapes and as such is not set in a real world context. There are ways that it could be adapted to appeal to the interests and experiences of your students. For example, students could be given the opportunity to decorate a model of their favourite polyhedron in a style of their choosing for a class display. This could range from cultural motifs to favourite colours, patterns or images.

Required Resource Materials
Activity

Session One

  1. Show the students slide one of PowerPoint One, which has all five Platonic Solids.
    What do you notice about these solids?
    What shapes make up each solid? (Platonic solids are made from one type of regular polygon)
    What are the flat surfaces (polygons) that make up the solid called? (faces)
    How many faces does each solid have? How did you count the faces systematically?
    Look for edges and corners (vertices – singular is vertex). How many edges and vertices has each solid got?
     
  2. The icosahedron is the most complex Platonic solid. From slide two of PowerPoint One students might be able to count the number of vertices, 3 + 6 + 3 = 12. If possible, have a model of the icosahedron available (made from interlocking shapes).
     
  3. Ask your students to imagine the model separated out.
    How many triangles will there be? (Twenty since icosa- is the prefix for twenty)
    How many corners (vertices) will the twenty triangles have in total? (20 x 3 = 60)
    How many of the sixty corners form each vertex of the icosahedron? (Refer your students to the model and systematically count the five triangles that surround each vertex)
    If five triangle corners make one vertex, how many vertices must the icosahedron have? (60 ÷ 5 = 12)
    Could the same kind of thinking help us to work out the number of edges? (The twenty triangles have 20 x 3 = 60 sides. Two sides are needed to make one edge. Since 60 ÷2 = 30 the icosahedron must have 30 edges.)
     
  4. By referring only to slide one of PowerPoint One ask your students to make models of all five Platonic Solids using interlocking shapes. Look to see that students understand that the arrangement of polygons around a vertex is consistent in each Platonic Solid (For example, a dodecahedron has three pentagons around every vertex). Ask the students to complete Copymaster 1 which systematically lists the number of faces, edges and vertices. Students are heavily guided on the copymaster to reinvent Euler’s famous theorem about networks (V + F = E + 2 where V equals the number of vertices, F equals the number of faces, and E equals the number of edges).
     
  5. Slide three of PowerPoint One shows a postage stamp featuring Leonhard Euler, commonly regarded as one of the three greatest mathematicians of all time. Students may want to research how Euler came to invent network theory as a branch of mathematics.

Session Two

In this session students work with truncations of the Platonic Solids. A truncation occurs when the vertices are ‘cut off’. PowerPoint Two begins with the simplest truncation, that of a cube.

  1. Look at Slide One.
    These two solids are related yet they look so different. How are they related?
    Students might notice if the corners of a cube are cut off the result is the right-hand solid, called a truncated cube. You can demonstrate truncation using a cube cut from a large potato or kumara. Where vertices are truncated new triangular faces are formed.


    How many faces, edges and vertices will the truncated cube have?
    What shapes are the faces? Why are the faces those shapes.
     
  2. Expect your students to use structure as well as visualisation to answer the question. For example, there are eight vertices on a cube so changing each vertex into a triangular face adds eight new faces. That gives a total of 6 + 8 = 14 faces. The faces made from a vertex are triangles because three squares meet at each vertex of a cube. That also means that 8 x 3 = 24 new edges are added making 12 + 24 = 36 edges in total. Each of the eight vertices of a cube are replaced by three new vertices. That makes 8 x 3 = 24 vertices.
    A truncated cube has 14 faces, 24 vertices and 36 edges. Does it still fit Euler’s theorem?
     

  3. Show the students slide two of PowerPoint Two. Ask the students to name the Platonic Solids (Tetrahedron and Octahedron) and recall the properties of the solids, including numbers of faces, vertices and edges.
    Here is the challenge. Use the connecting shapes to build the truncations of these solids. Before you start, visualise what the truncated solid will look like. Then build it.
     

  4. Give your students ample time to construct the solids. Look for the following:

    • Do your students apply structure to anticipate the result of truncation? For example, a tetrahedron has four vertices so four new faces will be formed by truncating each vertex. For the octahedron six new faces will be formed.

    • Do your students predict the shape of the new faces from the number of triangles around each vertex of the original solid? The tetrahedron has three triangles around each vertex but an octahedron has four. Therefore, the new faces will be triangles for a tetrahedron and squares for an octahedron.

    • Can your students anticipate how the original faces will be changed? Cutting off the vertices of triangles results in hexagons.

       

  5. Students can check their models against the picture on slide three of PowerPoint Two. Draw their attention to the arrangement of shapes around each vertex. For the truncated tetrahedron, one triangle and two hexagons surround a vertex. For the truncated octahedron, the arrangement is one square and two hexagons.
     

  6. Challenge your students further.
    Here are the dodecahedron and icosahedron. Imagine these solids are truncated.
    What shapes would the faces of the truncated solid be?
    How many faces would there be?
     

  7. Let your students solve the problem in small groups. Access to the models they built in Session One will be helpful. To build a truncated dodecahedron requires 20 triangles and 12 decagons (ten sided polygons). Therefore, a model cannot be built with a normal set of connecting shapes. However, a truncated icosahedron can be built from 12 pentagons and 20 hexagons.
     

  8. Slide four of PowerPoint Two has images of both truncated solids. Ask students to name the arrangement of shapes around each vertex. That arrangement is consistent for the whole solid.
    It is hard to visualise how many edges and vertices each solid has. You do have Euler’s theorem to help. Use your knowledge of the faces to work the numbers out.
     

  9. Look for your students to use the properties of shapes meeting at vertices and edges to solve the problem. They could organise the data in a table:

    Solid

    Number of faces

    Number of edges

    Number of vertices

    Truncated Dodecahedron

    20 triangles

    + 12 decagons = 32

    (20 x 3) + (12 x 10) ÷ 2 = 90

    20 x 3 = 60

    Truncated Icosahedron

    12 pentagons

    + 20 hexagons = 32

    (12 x 5) + (20 x 6) ÷ 2

    = 90

    12 x 5 = 60

    Does Euler’s theorem hold for both solids?

Session Three

In this session students create nets for the Platonic solids and possibly their truncations.

  1. Begin with this challenge:
    Use protractors, rulers and scissor to make and cut out an equilateral triangle, a square, a regular pentagon, a regular hexagon, a regular octagon and a regular decagon. Every side must be 5cm long. Use light cardboard.
    The purpose of making the shapes is to create templates to form nets with. Some students will need support with creating the polygons. Ideally students will use the sum of internal angles to work out the angle measures. That is a nice investigation but may interfere with the flow of this unit.

    Number of sides

    Name of polygon

    Sum of internal angles

    Each internal angle

    3

    Equilateral triangle

    180°

    60°

    4

    Square

    360°

    90°

    5

    Regular pentagon

    540°

    108°

    6

    Regular hexagon

    720°

    120°

    7

    Regular heptagon

    900°

    128.57°

    8

    Regular octagon

    1080°

    135°

    9

    Regular nonagon

    1260°

    140°

    10

    Regular decagon

    1440°

    144°

  2. For students who do not know the pattern you might provide Copymaster 2 which has the regular polygons on it. Students can measure the angles and use that information to create cut outs.


     
  3. Once the students have made cardboard polygons they can use them to construct nets for the Platonic solids and the truncations. Models made from interlocking shapes can be ‘unpeeled’, if needs be, to reveal a net that will work. By tracing around the shapes students can create nets quickly.
     
  4. The tetrahedron and cube are easy constructions. For the other three Platonic solids encourage your students to consider making the net for one half of the solid and joining two halves to make the complete net.

    The halves can be joined to form the full net.

     
  5. Of the truncated solids the tetrahedron and the cube are easiest. Encourage students to connect nets for the original Platonic solids with the nets for the truncated solids. For example:

     
  6. A net for the truncated cube can be made by a similar process.

Session Four

In this session students find the properties of shapes surrounding a vertex that can be used to predict whether a polyhedron will be formed.

  1. Set up the investigation as follows. Open up models of the Platonic solids to create nets.

     
  2. For each net find a vertex where two sides will fold up to form an edge of the solid. The black dots give examples of such points. The ‘missing angle’ is known as the angle defect. Slide One of PowerPoint Three gives the case of the octahedron.
    What is the angle defect, that is the angle that is missing from a full turn of 360°?
     
  3. Students should see that four angles of 60° exist at the vertex. Since 4 x 60 = 240 and 360 – 240 = 120, the angle defect is 120°. Some students may see that two equilateral triangles could fill the angle space so the defect equals 2 x 60 = 120°. The PowerPoint slide introduces a protractor so the analytical answer can be checked by measurement.
    How many vertices does the octahedron have? (six)
    There are six vertices where the angle defect is 120°.
    What is the total of six defects of 120 degrees? (6 x 120 = 720°)
     
  4. Slide Two looks at the cube. The defect angle is 90°. A cube has eight vertices, so the total of the defect angles is 8 x 90 = 720°.
    Maybe that is just co-incidence. Check out the total of the angle defects of the other Platonic solids.
     
  5. Let your students explore the other Platonic solids using the nets for support. Expect them to record the data systematically. Slide Three shows the angle defects and multiplies each defect by the number of vertices for the relevant solid.
    The students should notice that the sum of the angle defects is always 720°.
     
  6. Ask: How could we use this theorem to see if an arrangement of shapes will make a closed solid?
    Could we then know how many vertices the solid will have?
     
  7. Slides Four to Six have some arrangements of shapes around one vertex. The notation represents the regular polygons around each vertex. For example, (8, 8, 3) represents two octagons and one triangle about a vertex. Discuss whether they believe each arrangement will work. On each slide the truncated solid that matches the arrangement appears on the last mouse click.
     
  8. For each slide ask students to calculate the angle defect. If the arrangement works, then 720° is a multiple of the angle defect. With Slide Four the angle defect is 60° since the angles present add to 90 + 60 + 90 + 60 = 300. 12 x 60 = 720 so the arrangement will create a closed solid and the number of vertices will be twelve.
    The truncated cube has an angle defect of 30°, and 24 x 30 = 720, so the solid will have 24 vertices.
    The truncated octahedron also has an angle defect of 30°, and 24 x 30 = 720, so the solid will have 24 vertices.
     
  9. Slide Seven provides a final challenge. The arrangement of polygons has an angle defect of 360 – (108 + 60 + 108 + 60) = 24°. Since 720 ÷ 24 = 30 the arrangement will produce a polyhedron called the icosa-dodecahedron that has 30 vertices. It has that name because the centre of each pentagonal face is the vertex of an icosahedron, and in the centre of each triangular face is the vertex of a dodecahedron. Challenge your students to anticipate how many of each shape, triangle and pentagon, will be needed (20 triangles and 12 pentagons) before they build it.

Session Five

In this session students explore the classes of solids known as prisms and pyramids. Assigning solids to these classes allows the students to generalise the structure of nets and volume formulae. In the process a cylinder can be regarded as the ‘limiting case’ of a prism and a cone as the ‘limiting case’ of a pyramid.

  1. Ask students to make a triangular prism, a cuboid, and a hexagonal prism from the connecting shapes.
    What is the same about each solid?
    What differences are there among the solids?
     
  2. Look for students to talk about the defining feature of prisms, consistent cross-section if the solid is cut parallel to the ‘end’ faces. These faces name the prism, for example a triangular prism has a triangular cross section.
     
  3. Ask your students to sketch nets (flat patterns) for the prisms. The sketches can be confirmed by unpeeling the models you have. Standard nets for the prisms look like this.

    How do the similarities among the prisms show in their nets?
    How do the differences show?
    Students should note that all three nets have rectangular faces, and the number of those faces matches the number of sides of the end faces. The endpoint faces are those that give the prism its cross-section.
     
  4. Show the students Slide One of PowerPoint Four which is a cylinder.
    Is this solid a type of prism?
    Most definitions of a cylinder classify it as a curved surface. A prism is a type of polyhedron which means it is bounded (enclosed) by flat polygons. However, it is advantageous to consider the similarity of a cylinder to a type of prism. A cylinder has consistent cross-section (a circle) so the volume is worked out identically to any prism. Ask the students to sketch the net for a cylinder. The standard net looks like this:

    The net has much in common with the other nets for prisms. The two end faces are circles and there are an infinite number of infinitely small rectangular faces which form a continuous whole. The length of the whole rectangle is equal to the circumference of the circle cross-section.
     
  5. Ask: How do you find the volume of a cuboid? (rectangular prism)
    Students are likely to suggest length x width x height. Rebuild the cuboid model and ask how many 1 cm3 place value blocks will fit in. Highlight that multiplying two dimensions is like finding the area of the cross-section. Multiplying by the other dimension layers the cross-section.
     
  6. Ask: How do you think we could find the volume of the triangular and hexagonal prisms, and the cylinder?
    The same method applies. Find the area of an end face (triangle, hexagon, and circle) and multiply that area by the other dimension. Students may know the formula for the area of a circle a = πr2, though that knowledge is not expected at Level Four. That means the volume of a cylinder involves the area of the circular face multiplied by height (v = πr2 x h). You might test the formula out with a cylinder, e.g. a tin can. Measure the radius and height in centimetres and calculate the volume in cubic centimetres. Fill the container with water and measure to see if the capacity in millilitres matches the volume, since 1 cm3 = 1 mL.
     
  7. Follow a similar process with pyramids. Begin with three pyramids, triangular, square and pentagonal-based. Ask what is in common with these solids and what is different.
    Expects students to notice a base and triangular faces converging to an apex. Students should also note that the bases are different. Like prisms, pyramids are named by their bases, e.g. square based pyramid.
     
  8. Ask students to sketch the nets for these solids. Most students will provide the ‘flower’ shaped net which is the standard template.

     
  9. Students should be able to extend the idea to the net for a cone, though the curved surface of the solid is harder to visualise. Let students experiment to find out what pattern works. Notice that the flower shaped net does not look like the net for the cone since the curved surface is a single part circle (sector). However if all the triangular faces are joined in the nets above to easy to see that the limiting case, the number of sides in the base increases, creates a net with similar features. The arc length of the sector must match the circumference of the circle.
  10. The volume of a pyramid is related to that of a prism. Students might look the formula up and discover that volume of a pyramid is one third of the surrounding prism. For example, the volume of a cylinder is given by v = πr2 x h so the volume of a cone is given by v = ⅓πr2 x h.

    If you are ambitious, you might test the volume formula out by making a cone with card, lining the cone with plastic wrap and filling it with water.

Getting partial: Multiplying decimals

Purpose

This unit requires students to apply their number sense about the size of decimals to estimate and calculate the product of decimal fractions. In doing so they generalise about the effect of multiplying and dividing by ten and one hundred.

Achievement Objectives
NA4-3: Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.
Specific Learning Outcomes
  • Express a multiplication of two decimals as a product of fractions, e.g. 0.4 x 0.7 as 4/10 x 7/10 = 28/100.
  • Connect the product of the two fractions to the decimal answer, e.g. 28/100 = 0.28.
  • Know the effect of multiplying and dividing a decimal number by ten or one hundred.
  • Use multiplication and division by ten or one hundred to find the size of a product of decimals, e.g. 0.4 x 0.7 is the product of 4 x 7 divided by one hundred.
  • Use known benchmarks, especially one half and one, to estimate the products of two decimals, e.g. 0.4 x 0.7 is a bit less than 0.5 x 0.7.
Description of Mathematics

In this unit the students will develop number sense related to multiplying decimals. Making sensible estimates for the products of decimals requires a flexible connection of place number understanding with whole numbers, the meaning of multiplication, and multiplication with fractions.

In the age of calculators, it is a dubious use of instruction time to concentrate extensively on pencil and paper algorithms. Rather, emphasis should go on giving students strategies to recognise the reasonableness of answers and to estimate the products of decimals.

The decimal system includes a restricted set of equivalent fractions for situations where one units are inadequate for purpose. Commonly situations where tenths, hundredths, thousandths, etc. of units are needed involve measurement. The prefixes deci (tenth), centi (hundredth) and milli (thousandth) are applied to base units, such as metres and litres, to obtain a necessary degree of precision. Tenths, hundredths, thousandths, etc. are the result of division by ten of the previous larger unit. Therefore, understanding the effect of multiplying and dividing a decimal amount by ten, one hundred, and one thousand is foundational to estimating the products of decimals. The effect of division by ten is to make each unit one tenths of its previous size, represented as a shift of the digits one place to the right. For example, 3.9 ÷ 10 = 0.39, 3.9 ÷ 100 = 0.039, 3.9 ÷ 1000 = 0.0039.

Applying multiplication to measurement situations often involves a multiplier and a rate. A rate is a relationship between two measures, such as 60 kilometres per hour (velocity), 456 kilograms per cubic metre (density), or 30 people per square kilometre (also density). The multiplier is applied to the rate resulting in a measure, e.g. 3 kilograms of meat at $12.50 per kilogram costs $37.50.

Decimals might more correctly be termed decimal fractions. Deci is the prefix meaning one tenth, to indicate that decimal fractions are powers of one tenth, or negative powers of ten, e.g. 1/1000 = (1/10)³  or 10⁻³. Estimation of the size of products requires understanding of the multiplication of decimal fractions. Knowing 1/10 x 1/10 = 1/100 and 1/10 x 1/100 = 1/1000, etc., makes it possible to know the size of the products for factors like 0.6 x 0.4 and 0.3 x 0.08. Expressing both multiplications as fractions gives  6/10 x 4/10= 24/100 and 3/10 x 8/100 = 24/1000, so the products are 0.24 and 0.024 respectively. A sound understanding of multiplication of fractions is therefore a prerequisite for multiplying decimals with a sense of the size of product.

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 physical objects to connect decimals, as numbers, with physicial quantities. An area model is used extensively in the unit. You may need to revisit the area(array) model in simple whole number contexts to remind students about how the factors are side lengths and the products are areas.
  • Directly model the recording of equations, highlighting the fraction meaning of decimals and the symbols, x as ‘of’, and = as ‘the same as.’ For example, 0.3 x 0.9 = means “What is three tenths of nine tenths?”
  • Use calculators to confirm predictions about the results of multiplying decimals. Where differences between predictions and outcomes occur, ask students to work out where their predictions went awry. This strategy helps to cause cognitive conflict and address common misconceptions. The area model is particularly good in identifying where the size of units is misinterpreted.
  • Encourage students to work collaboratively in partnerships. Students need time to develop mathematical arguments and to rehearse those arguments with a peer is important for developing clarity and risk taking.
  • At first use rates that students are more familiar with, such as items in a packet, or speed in kilometres per hour. Progress to more unfamiliar rates such as density and fuel consumption.
  • Fold in an out of different levels of abstraction, i.e. materials, images (diagrams), equations. Use symbols as a means to connect across situations. Thinking with patterns in equations, and reasoning why patterns occur, is at the heart of mathematics.

The contexts for this unit are about rates, which is unavoidable with multiplication. Situations involve electricity and fuel consumption, density, and cost. Adapt the contexts if needed to meet the interests and backgrounds of your students. For example, using lengths of fabric at a decimal amount per metre, the cost of petrol per litre, or costs of vegetables, seafood, or other ingredient per kilogram, might be more familiar to your students. For students who have whānau in other nations, such as India or Samoa, currency exchange rates might be of interest.

Required Resource Materials
Activity

Session One

In this session students are introduced to multiplying decimals in everyday contexts. They refresh their understanding of multiplying fractions and the size of decimals using an area model. Students should work through the units Getting Partial to Fractions and Getting Partial to Decimals before attempting this unit.

  1. Begin with the first three slides of PowerPoint One. For each scenario:
    What operation do we need to carry out to solve this problem?
    Treat the significant digits as whole numbers to calculate the product:
    451 x 100 = 45 100              45 x 208 = 9 360                  15 x 537 = 8 055
  2. Ask your students to choose which of the four options is correct. Expect them to justify their choices using sense of the magnitude of the numbers. For example:
    • Slide One: If you get 4.5 Yuan for $1.00 then you should get 45 Yuan for $10.00 and 450 Yuan for $100.
    • Slide Two: 4.5 equals four and one half. 4½ x 2 = 10 so the answer must be close to $10.00.
    • Slide Three: 0.537kg is a bit more than half of one kilogram. The answer should be about half of $15.00 which is $7.50.
  3. Point out that the unit is about multiplying decimals and that number sense will be important in determining whether an answer is reasonable or not. Write the calculation 0.4 x 0.6 = ? on the board.
  4. Ask for the meaning of the symbols:
    What does 0.4 mean? (four tenths)
    What does 0.6 mean? (six tenths)
    What does x mean? (of)
    What does equals mean? (the same amount as)
  5. Ask your students to predict the answer to 0.4 x 0.6 = ?  Many may opt for 2.4 since both factors have the decimal point in the ‘middle’, as does 2.4.
  6. Slide Four of PowerPoint One shows how an area model can be used to find the answer. The model is important for understanding the metric units for length, area, and later volume. Four tenths of six tenths is shown by the orange rectangle. One unit of the 24 that fill the rectangle is shown.
    How big is the area that represents 0.4 x 0.6? (24 hundredths)
    What is the answer to 0.4 x 0.6? (0.24)
  7. You may have to put the digits of 0.24 on a place value chart to remind students of the nested nature of place value. In this case 20 hundredths are nested in 2 tenths. Slides 5 and 6 present two other examples of simple multiplication of decimals. In both cases challenge your students to predict the answers first before using the area model to confirm and make sense of the correct answer. You might provide Copymaster One to enable students to draw their own model of the multiplication.
  8. Provide Copymaster Two for students to practise solving simple decimal multiplication problems. Look for your students to:
    • Use the appropriate whole number multiplication fact.
    • Convert the decimals to fractions to mark the correct side lengths.
    • Understand that when tenths of tenths are found the answer is in hundredths.
    • Retain reference to one to find the answer.
    • Correctly express the answer as a decimal.

Session Two

In the next two sessions student learn more about the place value structure of decimals. They connect their knowledge of multiplication with fractions to finding decimals place value units, especially tenths of tenths, tenths of hundredths, and hundredths of hundredths. The learn about the effect of multiplying and dividing a number by ten and one hundred.

  1. Make several square metres from butchers’ paper.
    Who can mark one tenth of one tenth on this square metre? (You might use a place value block flat to mark the area)
    What will you call one tenth of one tenth? (one hundredth of a square metre(m²))
    Note that one hundredth of a square metre is a square decimetre (dm²) though the unit is rarely used in New Zealand.
    How might we record finding one tenth of one tenth? (1/10 x 1/10 = 1/100  or 0.1 x 0.1 = 0.01)
  2. Recall that last session you worked out calculations like 0.6 x 0.5 = ? 
    How do I write that calculation using fractions? (6/10 x 5/10 = 30/100 )

    Why is the answer not 0.030? (Thirty hundredths equals three tenths)
  3. Proceed in the same way to find:
    • One tenth of one hundredth equals one thousandth (1/10 x 1/100 = 1/1000  or 0.1 x 0.01 = 0.001) that can be marked with a place value long unit (1cm x 10cm).
    • One hundredth of one hundredth equals one ten thousandth (1/100 x 1/100 = 1/10000, or 0.01 x 0.01 = 0.0001) that can be marked with the area of a unit place value cube (1cm x 1cm). Note that one ten thousandth of one square metre is one square centimetre (1 cm²).
  4. Ask your students to recreate these units using the squares on Copymaster One. They should label the diagrams with the operation and the unit itself, e.g. 1/10 x 1/100 = 1/1000 ).
  5. Once students return, ask problems like:
    • If 0.6 x 0.05 = 6/10 x 5/100, why is the answer not 0.0030?
    • If 0.12 x 0.12 = 12/100 x 12/100, what is the answer? Why?
    • If 2.5 x 0.4 = 25/10 x 4/10, What is the answer? Why?
  6. You might take considerable time over the previous examples, asking students to draw diagrams to support their thinking.
  7. Provide students with Copymaster Three that deals with the patterns created when a number is multiplied or divided by ten. Students can attempt the problems in pairs with a calculator shared between them.
  8. Discuss a couple of the patterns to see if students correctly identify changes and consistencies between consecutive equations. See if they have generalised the effect of multiplying and dividing by ten.
  9. Watch Videos One, Two, Three, and Four to see why numbers behave the way they do when multiplied and divided by ten. For each video, ask your students to anticipate what will happen.
    What is the effect of multiplying/dividing by ten?
  10. Pose these problems for students to solve in pairs:
    3.7 x 10 =         3.7 x 100 =       3.7 ÷ 10 =        3.7 ÷ 10 =         1/10 x 3.7=        1/100 x 3.7=
    Be aware that students may not see that 3.7 divided by ten has the same result as one tenth of 3.7. That is an important connection if they are to use the effect to estimate the results of decimal multiplication.

Session Three

  1. Begin with PowerPoint Three as a guide to applying decimal multiplication to a real-life context. You might discuss density which is the ratio of mass to volume. Bringing in several objects with varying densities will help illustrate the attribute. For example, a basketball and a block of wood may look to have the same volume but vary considerably in mass.
  2. Work your way through the slides, inviting students to solve the problems. Do your students:
    • Have a reasonable expectation of the number size of each answer, using benchmarks like 0.5 as one half?
    • Understand that decimals can be thought of as fractions, to establish the size of the answer?
    • Recognise that division by ten can be used to find the answer in decimal form?
  3. Copymaster Four contains a set of problems that require multiplication of decimals. Let your students solve the problems in small teams. Most of the problems involve a rate, such as kilowatts per hour or litres per 100 kilometres. Rates in real life tend to be less familiar to students than the simple rates they encounter in multiplication normally, e.g. marbles per bag. After students attempt the problems it is worthwhile to discuss the rates that appear.

Session Four

This session is devoted to applying multiplication of decimals. The beginning explicitly confronts common misconceptions about decimals as a window to the understandings that students have developed so far. The final part of the session involves a game where multiplication of decimals is required.

  1. Work your way through the four slides of PowerPoint Four. On each slide challenge the students:
    • Is this student correct or incorrect?
    • Explain why they are correct or incorrect.
    • Suggest something to the student that will improve their understanding of decimals.
  2. Focus on the following issues in your discussion:
    • Slide One: ‘Adding zeros’ is a rule that students commonly learn. When a number is multiplied by ten, all the place value units become ten times the size. So effectively the digits move one place to the left, relative to the decimal point. The correct answer is 10 x 2.8 = 28.
    • Slide Two: Thinking that the decimal point is a separator can result in students operating of both sides of the decimal point independently. The role of the decimal point is to mark the ones place. The decimal places are connected. Felix could think that the answer is one tenth of the product of 6 x 48 = 288, so is 28.8. He could also use fraction multiplication 6 x 48/10 = 288/10 = 28 8/10 .
    • Slide Three: It is common for students to match the number of decimal places in the product with the number of places in the factors. Len could think that the answer is one tenth of one tenth (1/100) of the product of 8 x 3 = 24, so is 0.24. He could also use fraction multiplication 8/10 x 3/10 = 24/100, or 0.24.
    • Slide Four: Emma has generalised that zeros to the right do not affect the size of the decimal. For example, she thinks 3.70 has the same value as 3.7. In most cases that is true except when the zero on 3.70 represents a degree of accuracy, such as giving the number of centimetres, rounded to the nearest centimetre. Her answer is correct, possibly by accidently using Len’s incorrect idea. Emma should know that 1.5 is the same as one and one half. The answer should be one and one half lots of eight tenths, which is twelve tenths. So the correct answer should be 1.2 .
  3. After the class discussion introduce the game Decimal Pathways (Copymaster Five). The rules are as follows:
  4. Each player needs a pen of a different colour to their opponent.
  5. One copy of the game board is needed for each game. The gameboard could be laminated and whiteboard pens used instead of felt pens.
    • To Play
      Players take turns to:
      • Connect one factor on the left of their panel with one factor on the right, e.g. 0.3 and 0.8.
      • Calculate the product of the factors, e.g. 0.3 x 0.8 = 0.24.
      • Draw around the circle on the gameboard that has that product. If that product is not on the board the player misses that turn.
    • Once a circle is coloured it cannot be claimed by the other player.
    • The object of the game is to create a pathway of circles from a Player’s Start to their Finish. The first player to do that wins. If no player can make a pathway then the game is a draw. Note the pathway can be constructed in any order.
    • Look for your students to:
      • Calculate the products fluently using a combination of basic facts and place value knowledge.
      • Play strategically by capturing the circles in the centre of the board first.
      • Think ahead to capture circles that are most advantageous.

Cubic Conundrums

Purpose

This unit provides a set of learning tasks that integrate across the strands of the Mathematics and Statistics learning area of the New Zealand Curriculum and provide opportunities for assessment of student achievement across those strands. Each session may involve more than one lesson, especially if students need more time to become proficient at the tasks.

Achievement Objectives
GM4-3: Use side or edge lengths to find the perimeters and areas of rectangles, parallelograms, and triangles and the volumes of cuboids.
GM4-6: Relate three-dimensional models to two-dimensional representations, and vice versa.
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

Session One
Use systematic approaches to find all the possible outcomes, e.g. tree diagrams, organised lists.

Session Two
Use tables, graphs, and word rules to represent growing patterns.

Session Three
Draw cube models using plan views.

Session Four
Draw cube models using isometric projections.

Session Five
Find the volume of cuboids given edge lengths.

Description of Mathematics

The mathematics in this unit spans all three strands of the Mathematics and Statistics learning area of the New Zealand Curriculum. Key ideas are:

  1. Theoretical models of probability require knowledge of all the possible outcomes, or considerable data where those outcomes are not countable. In some ‘controlled’ situations it is possible to list all the outcomes using systematic strategies, like drawing tree diagrams.
  2. Growing patterns are one way to support the learning of relations, mappings between values for one variable with values of another variable. Attending to the spatial structure of patterns can support students to notice those properties of the shapes in the pattern that change and those that stay constant.
  3. Three-dimensional models exist in the real world. However, most representations of three-dimensional objects are two dimensional (flat). Some information about the real world object is lost in representing it two-dimensionally, while some information is retained. Different ways to represent objects, e.g. plan and isometric views, have nuanced ways to convey information about the object, e.g. square faces show as rhombi in isometric drawings.
  4. Volume is the amount of space contained in a bounded space, such as a packet or container. Cubes are the conventional unit for measuring volume as they ‘tessellate’ three-dimensionally with no gaps or overlaps. The arrangement of cubes in arrays allows for volume to be calculated by multiplication of edge lengths. However, partial units and curved surfaces require reorganisation of students’ expectations of getting discrete numbers of cubic units as measures of volume.

Specific Teaching Points

  1. Several systematic strategies can be used to count all the possible outcomes of a ‘controlled’ situation. These strategies include tree diagrams, tables, networks, and organised lists. The full set of possible outcomes can be used to express the probability of a chosen outcome as a fraction or percentage. There are six possible outcomes from rolling a standard dice and three of those outcomes are even numbers. The probability of rolling an even number is measured by 3 out of 6 or ½ or 50% or 0.5. Note that the ‘whole’ (one) is the set of all possible outcomes.
  2. Growing patterns can be investigated in three different ways:
    • Looking for structure in one example of that pattern
    • Looking at what changes from term to term as the pattern grows
    • Connecting representations of the pattern, to deduce the properties of any term Tables, graphs, and rules or equations are common ways to represent the relationship between variables (quantities that change) that are found in growing patterns.
  3. Plan views of buildings represent three dimensional objects in flat space (two dimensions). As plan views do not show depth, information from several views must be co-ordinated to gain a more complete understanding of the structure of the building. Similarly, isometric drawings display some information about depth but features of the building can be masked behind other features.
  4. Recognition of the array structure that underpins calculation of area and volume is problematics for many students. Experience with filling spaces is vital for students to understand why squares are the conventional unit for area and cubes are the conventional unit for volume. The ‘no gaps or overlaps’ and ‘identical size’ properties of units are important yet often not obvious to students. Multiplication of side and edge lengths to find area and volume is based on the systematic arrangement of units in arrays.

Observations of students during this unit can be used to inform judgments in relation to the Learning Progression Frameworks. Click for tables of guidelines.

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:

  • Provide opportunities for physical experience with cube models prior to asking for abstract thinking. Ask students to predict what will occur as a model is altered before they change the physical appearance, e.g. What will the fourth model look like?
  • Start with challenges the are in the ‘build’ zone first, before giving challenges that make physical building too cumbersome. For example, provide simple cube structure to draw first, moving to more complex structures later.
  • Provide supportive tools for students to represent and manage their ideas, for example, using tables for sequential pattern, and satellite views for plan drawings.
  • Use technology to shift student attendance onto more complex thinking, for example experimenting with direct rules in a spreadsheet or considering hidden cubes when making an isometric drawing digitally.

Activities can be adapted to meet the interests and cultures of your students. Where necessary use internet resources to support students to understand contexts. For example, students may not be familiar with apartment complexes, and architectural drawings. Find resources that broaden their world view. As much as possible, give students the opportunity to be creative mathematicians, for example, create their own cube structure to draw. Foster a climate of problem posing as well as problem solving. Contexts can also be altered, for example a townhouse of two colours might become a tātua, traditional woven belt, and a cube structure might be reframed as the contours of pa site. Arrangments of foods for storage, e.g. kumara in pits, might provide a nice context for volume.

Required Resource Materials
Activity

Session One: Two Colour Townhouses

In this session students explore combinations with two or more colours in a probability situation. They link the combinations to powers of two and to possible outcomes.

  1. Use PowerPoint One up to the end of Slide Two to introduce the “Townhouse Problem.” Get a couple of students to make two possible townhouses using two colours. Two important points are:
    • The ground sets a reference in this problem in terms of deciding whether or not two towers are the same. If the earth is considered as a reference these two townhouses are not the same:
    • Since colours in a set of blocks are limited use light and dark colours to create the contrast needed, e.g. black and white, blue and yellow, etc. This will allow more students to use the same set.
  2. Allow the students plenty of time to investigate the four-storey townhouse problem in small co-operative groups. They will need access to connecting cubes and squared grid paper will support them to record their solutions. Look for the following:
    • Are the students systematic in their approach or do they just create solutions without checking for uniqueness? If they are not systematic, ask the question, “How will you know when you have found them all?”
    • Do they simplify the task by working through sets of colours? i.e. Divide the search into 4 darks, 3 darks and 1 light, 2 darks and 2 lights, 1 dark and 3 lights, 4 lights.
    • Do they look for symmetry to find new combinations? For example, these new townhouses can be found by reflection (or rotation).
    • Do they notice structure in solutions? For example, there are only four possible positions for one dark cube so there must be four possible towers with 3 lights and 1 dark. That must also be true of the inverse image of 3 darks and 1 light.
  3. After a period of investigation bring the class together to discuss their solutions. Focus on the points above. Listen to the students’ methods first than use Slides 3-6 of the PowerPoint to discuss other possible strategies.
  4. A “Simpler Problem” strategy reveals there are two possibilities for the one-storey townhouse, four possibilities for the two-storey townhouse, and eight possibilities for the three-storey townhouse.
    How many possibilities might we expect for the four-storey townhouse?
  5. Slide 7 shows the data organised in a table. Do the students recognise that powers of two are involved? For example, 8 = 2 x 2 x 2 = 2³.
  6. Slide 8 shows the doubling of possibilities occurs as a new storey of two possible colours is added to the bottom. Each existing possibility has two possibilities for the next storey.
    • Can students draw the tree diagram for the four-storey houses so they find all 16 possibilities?
    • Can they match arms of the tree diagram to a model made from cubes? Other questions for investigation include:
    • How many five-storey townhouses would be possible with two colours of cladding?
    • How many three-storey townhouses would be possible with three colours of cladding?
    • How many four-storey townhouses would be possible with three colours of cladding?
    • Is there a rule to determine the number of possible townhouses with s storeys and c colours?
  7. Suppose the people who buy the townhouses select their colour scheme by reaching into a bin of cubes, with equal numbers of dark and light. The first colour out is the ground floor, then it is replaced in the bin. The second colour out is the second floor, then it is replaced in the bin, etc.
    What are the chances that the buyer will get a townhouse of only one colour? (two out of 16 or one-eighth)
    What are the chances that the buyer will get an alternate pattern?, i.e. dark-light-dark-light or light-dark-light-dark. (Also two out of 16 or one-eighth)
    What are the chances of the buyer getting two storeys of each colour? (six out of 16 or three eighths)
    Students might make up other possible colour options and work out the probability of that option occurring. They might act out the drawing cubes scenario to see what colour schemes they actually get with 16 attempts. Does every possibility come out? Why is this unlikely? High achievers might investigate the connection between the possible townhouses and Pascal’s triangle.

Session Two: Square Doughnuts

In this session students use connecting cubes to create growing patterns and attempt to find general rules for those patterns. They also graph the growing patterns using spreadsheets and look for patterns in the ordered pairs.

  1. Introduce the cube model using Slide One of PowerPoint Two. Ask the students to work out the number of cubes required. Highlight the importance of structuring the pattern in parts. Invite suggestions about how to count the cubes. Record the suggestions as equations. Slides 2-6 offer some ways to structure the pattern.
    • Slide Two: The person sees 4 x 4 + 4 = 20
    • Slide Three: The person sees 2 x 6 + 2 x 4 = 20
    • Slide Four: The person sees 4 x 5 = 20
    • Slide Five: The person sees 4 x 6 – 4 = 20 (Subtract four to allow for overlaps)
    • Slide Six: The person sees 6 x 6 – 4 x 4 = 20 or 6² – 4² = 20
    • Slide Seven shows three members of the growing pattern. Ask the students to make models of the pattern members using cubes. What changes and what stays constant (the same) as the pattern progresses? Students should notice that the side length is growing by one cube each successive figure to the right. That defines one variable, side length (in number of cubes). Students should make other observations like: 
      • The hollow square in the centre is also getting bigger. Each side length of the square grows by one cube each time.
      • The number of cubes needed to make each model is four more than for the model before it. Why does that happen? (One cube is added to each side of the model).
  2. Pose this problem to the students:
    Imagine someone gave you 96 cubes to build a model in this pattern. How long would the side length of your model be in cubes?
  3. Let the students solve the problem in whatever way they want. Look for the following:
    • The choice of some students will be to build the figure. Ask them to think about what they must consider as they make their model larger. They should note that all four sides must be the same length. Can they predict ahead to anticipate the side length of the complete 96-cube model?
    • Some students might use tables to organise their data. If their attendance is to the ‘going up by four’ recursive rule encourage them to think ahead without repeatedly adding. Ideally you would like them to find a relation between side length (s) and total number of cubes (c).
    • Some students will bring through the structure they used to count the number of cubes in the figure with side length of six. They will imagine what side length might get them close to 96 cubes. For example, a model with 20 cubes on each side has 4 x 19 = 76 cubes in total.
  4. After a suitable time of exploration gather the class to discuss their ideas. Try to connect different strategies with questions like:
    Why would the sequence going up in fours give you a rule that uses ‘multiply by four’?
    Would the problem be easier if you had 400 cubes instead? Why? What would you do to solve it?
  5. The downloadable spreadsheet has two pages; page one contains the start of a table of values for the pattern and page three has a lengthy table to n = 25. Start with page one and ask the students ideas about how to extend the table to solve the problem for 96 cubes.
    The problem can be solved by auto-filling but encourage students to suggest formulas that might be used. One possibility is to put the formula =A5+1 into cell A6 and =B5+4 into cell B6. These are recursive formulas that work out members of the sequence from the one before. Copying down the formulas into columns A and B will continue the pattern.
  6. Encourage the students to look for a relationship across the table between the values in columns A and B. Since the spatial pattern involved four sides, multiplying the A values by 4 is a good start. The final adjustment is to subtract 4 to allow for overlaps. So the formula =A6*4-4 in cell B6 will give the total number of cubes if 7 is entered into A6. This formula can be copied down to complete the table.
  7. Page two has a completed table and a graph of the relation. Ask the students what patterns they notice in the graph and why the patterns occur. They should notice that the points (ordered pairs) lie on a straight line. This occurs because the rate of change, four more cubes for each extra cube of side length, is constant. The rise is four for every run of one. You might ask students to create their own table and graph for the growing pattern using Excel.
  8. Use Slides 8-10 of PowerPoint Two to motivate the students to create their own growth patterns from connecting cubes. They need to build three members of the pattern, preferably consecutive members of the pattern. Next they need to pose a problem for someone else to solve. The problem could be:
    • Make the next model in the pattern
    • What would be the tenth model in the pattern?
    • What model in this pattern would you make with c cubes?
    • Create a spreadsheet and graph to predict the number of cubes needed for the 50th member of the pattern.
  9. If possible use digital cameras to take pictures of the models and use them to publish the students’ problems. If students do publish their problems they need to create a model answer along with useful hints for other students.

Session Three: Spylights

In this session students explore plan views to represent three dimensional models made with cubes. They interpret the plan diagrams and create their own diagrams for others to build from.

  1. Begin by building a model with cubes in the shape of an apartment block. Place the model on a desk and invite a student to draw what the model looks like from the front. If you have an electronic whiteboard have a square grid showing to make drawing easier and more consistent. An A3 sized sheet of Copymaster 1 will also work. Invite other students to draw the ‘Bird’s eye’ and left side (relative to the front) views.
  2. Use PowerPoint Three to introduce the problem of the three views. Draw students’ attention to these features of the plan views:
    1. What does each square represent? (A single face of a cube)
    2. How many cubes across is the building? (Four as shown by both the front view and Satellite image)
    3. What is height of the tallest part of the building? (Four cubes high as verified by both the front and left plans)
    4. How tall is this part of the building (pointing to the front square of the satellite image)? (Two cubes high at it is the only stack of cubes visible on the right of the left image).
  3. You might want to provide the students copies of Slide Three. Ask them to work in small groups of two or three to build a model that matches the plan views that Simon, the spy has taken. Look for:
    • Are the students systematic or do they attempt then dismantle their model and start again at the beginning if it does not work?
    • Do they start with one view first, e.g. Satellite image, get that view correct then progressively work on the other two views?
    • Do they check all three views to validate their solution meets all the requirements?
    • Do they use recording strategies to record the important information in the problem? For example, using the Satellite image and recording the maximum height of each row and column is useful.
  4. After a suitable period of solving the problem gather the class to share their strategies. Look for some students to create familiar chunks of the building to work with. They might create towers of heights 1, 2, 3, and 4 to match the eights they see in the front and left views, then move those towers around to match the Satellite image.
  5. Provide the students with copies of Copymaster 1. Ask them to create three views of a building they create. Restrict the students to 12 cubes so the problems become manageable. After they have drawn their views students can exchange their three view plans with another student, and ask that student to make their building. Digital cameras can also be used to create model answers of the target solid.
  6. Another good activity is to leave the buildings intact. Each student then receives a set of plans and must tour the room until they find a model that matches their plans.

Session Four

In this session students learn to represent models made with connecting cubes using isometric dot paper (orthogonal diagrams or projections). Students will need isometric dot paper (Copymaster 2) and connecting cubes.

  1. Begin with an isometric drawing beginner class using PowerPoint Four. Using mouse clicks on Slide One students can progressively draw some simple cube models. Show them Slide Two and invite them to make the model first then recreate the drawing of it. Students could also draw the building model they created in Session Three.
  2. After students have attempted drawing discuss some important features of isometric drawing:
    • Why is the page described as ‘isometric’? (‘Iso’ means the same, and ‘metric’ means measure. So the dots are all the same distance apart.)
    • What shape are the squares faces of a cube when drawn on isometric paper? (Non-square rhombi – four sides of equal length.)
    • How do you decide which faces to shade? (Imagine a light source and shaded the opposite faces to that source.)
    • Can you always see all the cubes? (No often parts of a cube or a whole cube are hidden behind others.)
  3. Slide Three of PowerPoint Four introduces the pieces that make up The Steinhouse Cube puzzle, invented in 1950 by a mathematician, Hugo Steinhouse. Ask the students to use the photograph to make all six pieces and draw them on isometric paper. Challenge students to solve the puzzle of putting the pieces together to form a 3 x 3 x 3 cube. Students might video their solution for others. Clips of people solving the puzzle are easily found by internet search.
  4. As a challenge students might like to make up their own cube puzzle, draw the pieces, and an ‘exploded’ solution on isometric paper showing how the pieces connect. If they are unsure what an exploded diagram looks like ask them to search online for examples. Exploded diagrams are important ways to provide assembly instructions for different products, from flatpack furniture to complex machines. Other isometric drawing challenges can be found in the following Figure It Out activities:

Session Five

In this session students learn to find the volume of a cuboid (rectangular prism) by visualising the arrangement of cubes that will fill it. They also connect multiplication of length, width and depth to the arrangement of cubes.

  1. Slides 1-5 of PowerPoint Five introduce the idea of finding the volume of a cardboard packet by filling it with cubes. The issue of part units is also raised as two edges of the packet measure 7cm or the length of 3½ cubes. Provide the students with an assortment of cuboid shaped boxes of varying sizes, e.g. Toothpaste, biscuit, cereal, shoe, etc. Tell the students that you want them to find the volume of each box in connecting cubes. Let the students work in groups of two or three. Expect them to record measures for the volume of each box and how they found that measure. For example, Weightwatchers muesli bar box, 7 cubes high by 2 cubes across = 14 cubes, 8 layers is 8 x 14 = 112 cubes.
  2. As the students work look for:
    • Do the students recognise that the cubes can be arranged in a 3-dimensional array?
    • Do the students recognise that a ‘layer’ can be created and mapped into the box as many times as needed to fill it?
    • Do the students apply multiplication to find the volume, or do they rely on counting or repeated addition?
    • Do they record the measures using both numbers and units?, e.g. 4 x 5 x 3 = 60 cubes.
    • How do the students deal with partial units? Do they work with fractions of a cube to get more precision?
  3. After a suitable period of exploration bring the class together to discuss the key points above. Slides 6-8 of the PowerPoint show how to work out the volume of a box from edge lengths. It also illustrates that two factors must be multiplied first. The first two factors represent a layer that is iterated (copied) through the box. For each slide represent the volume calculation with a multiplication equation, i.e. 4 x 3 x 5 = 60 (Slide 6), 5 x 3 x 4 = 60 (Slide 7), and 3 x 4 x 5 = 60 (Slide 8). Ask the students what they notice. The associative property of multiplication means that the pairing of factors does not affect the product.
  4. You might also look at cubic centimetres (cm³) the standard unit of volume. That is the size of a small place value block. Most likely the students will have measured their boxes using cubes that are 2cm x 2cm x 2cm.
    How many cm³ fit into one of the connecting cubes? (2 x 2 x 2 = 8). So if the number of cubes is multiplied by 8 then that gives the volume in cubic centimetres.
  5. If you return to Slides 1-5 the volume of the packet can be worked out more accurately by 20 x 7 x 7 = 980 cm³. This is close to 1 litre which 1000 cm³.
  6. Provide additional challenges like:
    How many different cuboids can be made with the cubes that have a volume of sixty 2 x 2 x 2 = 8cm³ cubes? Sixty cubes is a good target volume since 60 has many factors. Slides 6-8 give one possible answer of 3 x 4 x 5 = 60 cubes. Uniqueness is an issue. For example, in this investigation 4 x 3 x 5 should be considered as the same cuboid as 3 x 4 x 5.
    Do the students approach the task systematically?
  7. Ask: How will you know when you have found all the possibilities?
    One approach is to close off the first factor, like this:
    1 x Possibilities: 1 x 1 x 60, 1 x 2 x 30, 1 x 3 x 20, 1 x 4 x 15, 1 x 5 x 12, 1 x 6 x 10 (all done – How do you know?
    2 x Possibilities: 2 x 2 x 15, 2 x 3 x 10, 2 x 5 x 6 (Why we not include any ones as factors)
    3 x Possibilities: 3 x 4 x 5 (All done – How do you know?)
    Are there other possibilities? (No. Why not?)
  8. Other target volumes can lead to other discoveries. For example, a prime target volume such as 29 has only one solution 1 x 1 x 29. Why? A target volume that is a power of two will have only result in edge lengths that are powers of two, e.g. 64 cubes: 1 x 2 x 32, 2 x 2 x 16, 1 x 8 x 8, etc.

Printed from https://nzmaths.co.nz/user/75803/planning-space/late-level-4-plan-term-3 at 11:22am on the 21st January 2022