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.

Level Three

Integrated

Units of Work

This unit seeks to connect learning outcomes, and provide problem solving opportunities across all five content strands, number, geometry, statistics, algebra, and measurement. The context of houses is used to develop concepts such as drawing and modelling 3-dimensional objects, using co-ordinate...

Level Three

Number and Algebra

Units of Work

This unit builds students’ understanding of place value, extending to 6 digit whole numbers.

Level Three

Geometry and Measurement

Units of Work

In this unit students use street maps as the context to learn about coordinates and grid references, and for giving and following instructions involving directions and distances.

Level Three

Number and Algebra

This unit of work investigates how relational thinking is applied to multiplication and division.

Level Three

Statistics

Units of Work

In this unit we investigate the amount of water contained in a carrot and we use a time series graph to plot the "weight" of the carrot as the water in the carrot evaporates. The students also pose their own carrot investigative questions.

Level Three

Statistics

Units of Work

In this unit, using our fridge as the context, we collect data and present these as dot plots and bar graphs. We start to learn about using the computer to display our data.

Printed from https://nzmaths.co.nz/user/75803/planning-space/late-level-3-plan-term-1 at 12:03pm on the 21st January 2022

## Houses

This unit seeks to connect learning outcomes, and provide problem solving opportunities across all five content strands, number, geometry, statistics, algebra, and measurement. The context of houses is used to develop concepts such as drawing and modelling 3-dimensional objects, using co-ordinate systems to locate position, finding all the possibilities of events, and identifying paths through simple networks.

This multiple-strand unit covers a wide range of mathematics topics including: properties of triangles and quadrilaterals, networks, nets and diagrams of 3-dimensional solids, co-ordinates on a number plane, and combinations.

It supports students who are working within stage 6 (advanced additive) of the Number Framework. At stage 6 a student can estimate answers and solve addition and subtraction tasks involving whole numbers by choosing appropriately from a broad range of strategies (for example place value partitioning, rounding and compensating or reversibility). The student uses a combination of known facts and a limited range of strategies to derive answers to multiplication and division problems (for example doubling, rounding or reversibility).

This unit can be differentiated by varying the scaffolding provided to make the learning opportunities accessible to a range of learners. For example, drawing views of 3-dimensional structures can be scaffolded using variables like:

The practical nature of the tasks in the unit requires access to physical and diagrammatic resources. Providing materials and putting varied demands on what tasks are carried out mentally, is a powerful tool in catering for diversity.

The contexts in this unit can be adapted to recognise cultural and geographical diversity to encourage engagement. For example, historic home tours might become tours of the buried village outside Rotorua, or the Waimunga Valley pathways. Constructing nets for solids might be framed around models of whare nui or whare kai, or of significant buildings in the local area.

## Session 1

Show the students House Plan One (Copymaster 1). Illustrate how the house designer drew the plans by setting up an object on a desktop and drawing the object as seen from the front, side, and top as you move around the object. Provide the students with plasticine or modelling clay and ask them to make a small model of the house that is shown in these plan drawings.

After students have attempted this get them to check their model by sketching it on a desktop by moving around it to get the front, side, and top views. The drawings can be checked against the original plans. Discuss with the students what features they concentrated on in producing the model. This might include thinking of the model in two parts - the basic cuboid (rectangular prism) and the triangular prism that forms the roofline.

Provide the students with House Plan Two, from Copymaster 1, and challenge them to make a model of that house. Help scaffold the task by discussing what features of the house are different to the previous plan. Students may note that the new house has the same roofline but the top view shows that it is L-shaped. As with the first plan the models can be checked by using them to produce plan drawings.

The activity can be extended in two ways. Firstly tell the students that architects frequently make models of their buildings for presentation purposes. Get them to design the net (flat pattern) that could be folded up make a model of house one. This could be produced using the plasticine model by rolling it on a sheet of paper and tracing around each face as it lies against the paper until all the faces have been traced. Students should note that use of symmetry can make the process much easier.

One net of house plan one looks like this:

Discuss with the students how nets usually have tabs to allow them to be glued together. The usual convention is to put a tab on every second edge as you work around the perimeter (outside) of the net but other ways are also workable. Creating the net for House Plan Two is considerably more difficult but can be done in halves and symmetry used to create the full net.

Another extension of the activity is for students to make their own house model from plasticine and create front, side and top view drawings of it. These drawings are then given to another student who must produce either the model or net for the plan.

## Session 2

Ask several students in the class how far they live away from the school. They may give their answers in distance or travel time. Comment that they are going to investigate where the school should be to make it as central as possible. Take two students and ask them to mark as accurately as possible where their house is on a street map of the area. Ask the students if the school is central (in the middle) of these two houses. This is easy to visualise by drawing a line between the two locations:

Discuss the methods that students have used. These will be focused on features of the triangle, particularly equally splitting angles and sides. Such methods might produce solutions like:

Split the anglesmethod.Split the sidesmethod that, in this case, does not appear to give a sensible solution.Width by heightmethod. Draw the smallest rectangle that includes all of the corners and cut the sides in half. The intersection of these “bisectors” gives the centre.The co-ordinatemethod. Each house location is given an ordered pair and the average of the x-co-ordinates and the average of the y-co-ordinates is found. This gives the centroid or balance point of the triangle.The average of the co-ordinates gives the point (7, 4.33…).In this case the co-ordinate averaging method seems to give a good indication of the centre.

Discuss with the students which of these methods will also work if we consider finding the centre of four or more houses. They may conclude that the length by width and co-ordinate methods will always work, no matter the number of houses. The split the angles and sides methods could be tried with some quadrilaterals to determine if they will work.

Apply both the length by width and co-ordinate methods to the task of finding the central location for a school that only included students in your class. Use a scale map of the district. Draw a co-ordinate plane over the map and ask each student to provide the location of their house as an ordered pair. Note that decimal ordered pairs could be used to make the locations more accurate. A spread sheet might be used to find the average of all the co-ordinates quickly.

Name

Xco-ordinateYco-ordinateJoe

3.5

6.4

Gill

0.9

8.7

Shirley

6.5

1.2

…

…

=average(B2:B?)

=average(C2:C?)

Compare the centre as determined by both methods. Discuss which method appears to be the most accurate. Students may note that outlier (extreme) co-ordinates have a pronounced effect on the length by width method but minimal effect on the average co-ordinate method. This could be modelled by introducing a hypothetical student who lives an extreme distance form the school and noting the effect using the spreadsheet and map.

## Session 3

Tell the students that they are going to investigate the best way to build framing on the wall of a house. Using cardboard strips and split pin paper fasteners get them to construct the following “walls.”

Point out the need to accurately measure the strips and location of the holes so that the walls lie flat. Which one is more stable as the walls of house need to be reasonably rigid? The students should notice that the vertical and horizontal arrangement tilts while the triangular arrangement stays fixed.

Ask the students why they think this occurs. They may comment that triangles are strong while the angles of the rectangles are movable. Discuss why the triangular wall they have made, while very strong, might not be functional in building houses. This relates to the difficulty of measuring angles so that the walls will intersect and the difficulties of living in dwellings with non-perpendicular (right-angled) corners. A-framed houses are a good example of the later problem. Building a tetrahedron with polygons might be a good way to demonstrate the inefficiency of space usage (let alone trying to clean!).

To develop students’ ideas about the structural strength of triangles and quadrilaterals get them to carry out the following investigation.

Make the following lengths from cardboard strips. The lengths show the length between the holes:

Ask the students to make as many different triangles as they can by fastening the cardboard strips with fasteners and moving the sides around. They will realise that only one triangular shape is possible as the side lengths determine the angles at the corners.

Ask them to use another length of cardboard strip (for example 18cm between holes) and use it with the other lengths to make quadrilaterals of different shapes. The students will see that the angles at the corners can be changed by tilting the quadrilateral. This could be used at another time to investigate the sum of interior angles.

The students might try making triangles and quadrilaterals with different lengths to further validate their conjectures if need be.

Extend this idea by telling the students that they are allowed to add only one strip of card and fasten it to the original vertical and horizontal wall arrangement they started with. The challenge is to stabilise the wall so it does not tilt. This can be done by using a brace that effectively turns the quadrilaterals in the wall into triangles with rigid structural strength.

## Session 4

Tell the students that they have been hired by a construction company as a consultant to look at the colour schemes for a new housing estate (Copymaster 2). The houses all have the same design so the company is keen to use the colours to make each house appear different to the others in the estate.

The company has only bought three colours of paint red, blue and yellow. They have decided that for each house the roof will be painted in one colour, the walls in a different colour, and the window sashes and door in a third colour. They are wondering if three colours are enough. How many houses can be painted with three colours remembering that each scheme must be unique? That is, no two houses can look the same. Encourage the students to be systematic in planning the colour schemes. Provide them with copies of Copymaster 2.

After an appropriate time of investigation bring the students together to share their ideas. Focus on strategies they have used to systematically find all the possibilities. Strategies might include:

Using tablesRoof

Walls

Door/Windows

Red

Yellow

Blue

Red

Blue

Yellow

Yellow

Red

Blue

Yellow

Blue

Red

Blue

Yellow

Red

Blue

Red

Yellow

Using tree diagramsRoof Walls Door/Windows

Focusing on the structure of these models reveals that the six combinations are found by finding three possibilities for the roof colour multiplied by two possibilities for the walls (that is 3 x 2 x 1).

The company has built 120 houses in the estate so obviously three colours are not sufficient to make each colour scheme unique. Ask the students to predict how many colours will be needed to produce 120 unique colour schemes. Intuitively it seems that a large number of colours will be required.

Suggest to the students that they progress systematically in finding out the number of combinations when each new colour is added. This should involve “adding on” to the tables, diagrams, or other models that they used in the simpler example with three colours.

The students may notice that the addition of a fourth colour, say green, results in a four fold increase in the number of possibilities. Extending the table illustrates this clearly:

Roof

Walls

Door/Windows

Red

Yellow

Blue

Green

Red

Blue

Yellow

Green

Red

Green

Blue

Yellow

Yellow

Red

Blue

Green

Yellow

Blue

Red

Green

Yellow

Red

Blue

Blue

Yellow

Red

Green

Blue

Red

Yellow

Green

Blue

Green

Red

Yellow

Green

Red

Yellow

Blue

Green

Yellow

Red

Blue

Green

Blue

Red

Yellow

Similarly drawing the tree diagram will reveal four possibilities for the roof colour multiplied by three possibilities for the wall colour multiplied by two possibilities for the door/windows colour (4 x 3 x 2 = 24).

Students will have to look for some generalisation in determining how many possibilities occur with five colours. Creating all of these possibilities would be a painstaking process. Ask the students to discuss in groups what might happen when the fifth colour is used. Look for reasoning like:

Roof Walls Door/Windows

5 colours x 4 colours left x 3 colours left gives 60 possibilities.

This would mean that a sixth colour would create 6 x 5 x 4 = 120 possible colour schemes which would be enough to make each of the 120 houses unique.

Students might explore other scenarios that are either simpler or more complex. For example:

What happens to the number of possibilities if it is decided that the roof, windows, and doors are painted the same colour?What happens to the number of possibilities if the windows and door are painted in different colours?Students might also be asked to solve problems that are similar in structure. For example:

You have these digit cards in a pack: 0, 1, 2, 4. You deal out three cards. 4 then 2 then 0 come up.What other card combinations might have come up?

How many different combinations are there?

## Session 5

Discuss with the students why people might be interested in very old houses. Tell them that touring large, old houses is very popular with tourists. As the houses are very precious it is important to design tours that do not involve returning frequently to the same room or doorway. Tourists don’t like repeating themselves either.

Use an overhead projector to show the plan of the rooms in "Comfy Cottage" (Copymaster 3). The gaps indicate doorways and the start and finish indicate where the tour is to begin and end. Tell the students to mark a tour through Comfy Cottage. Note that there are several possible tours. One possible tour is given on Copymaster 3. Discuss strengths and limitations of the tours. The example given returns to the lounge.

Is it possible to tour the whole house without returning to the lounge?Tell the students that they are going to be given several historic homes to design tours for. They will need to decide which rooms to start and end the tour on and check to see whether the tour will go through each doorway only once. Provide the students with copies of Copymaster 4 and encourage them to work in pairs in solving the tour problems.

After an appropriate time bring the class together to share solutions. Concentrate firstly on how the students identified which rooms to begin and end the tour at. Look for common features of these rooms. Students might notice that these rooms always have either one or three doorways. Discuss what they notice about the rooms that are passed through. Focus on the number of doorways in these rooms. Students may notice that they have either two or four doorways.

Look at both scenarios, an odd or even number of doorways. For a room with an even number of doorways a tourist can either start inside or outside it.

If a tourist starts inside the room they must return to it.

If a tourist starts outside the room they must end up leaving it.

For a room with an odd number of doorways the opposite scenario is true.

If a tourist starts inside the room they can return to it many times but always end up leaving it.

If a tourist starts outside the room they can pass through it several times but will always end up returning to it.

In this analysis it is easy to see that any house cannot have more than two odd doorway rooms if the tour is to work. This means that the tour must start in one odd room and end in the other. Along the way, any even-doored rooms are entered then left.

Challenge the students to try this theory out by creating two house plans where a tour is possible and two plans where a tour is not. Students can share these plans to help confirm that a tour is only possible given no more than two odd-doored rooms.

## Place value with whole numbers

This unit builds students’ understanding of place value, extending to 6 digit whole numbers.

Understanding place value is crucial if students are to develop the estimation and calculation skills necessary to become numerate adults. Our number system is based on groupings of ten. Ten ones form one ten, ten tens form one hundred, ten hundreds form one thousand, etc. The system continues, giving us the capacity represent very large quantities. The place values, one, ten, one hundred, one thousand, etc., are powers of ten. That means that the place immediately to the left of a given place represents units that are tens time more than the given place, e.g. thousands are ten times greater than hundreds.

Ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used to represent all the numbers we could ever need. We don’t need a new number to represent ten because we think of it as one group of ten. Similarly, when we add one to 999, we write 1000 and do not need a separate symbol for one thousand. The position of the 1 in 1000 tells us about the value it represents. Zero has two uses in the number system, as the number for ‘none of something’, e.g. 6 + 0 = 6, and as a place holder, e.g. 7040. Place holder means it occupies a place, or several places, so the reader knows the values represented by the other digits. In 7040 zero is acting as a place holder in the hundreds and tens places.

Place value means that both the position of a digit as well as the value of that digit indicate what quantity it represents. In the number 2753 the position of the 7 is in the hundreds column which means that it represents seven hundred. Two is in the thousands column which means that it represents 2 units of one thousand, called 2000.

Understanding the nested nature of place value becomes very important as students learn to operate on whole numbers and extend their knowledge to decimals. Nested means that the places are connected, e.g. within hundreds there are tens, within ones there are tenths. Renaming a number flexibly is an important application of nested place value.

In particular it is vital that students understand that when ten ones are combined they form a unit of ten, when ten tens are combined they form a unit of one hundred, and when ten hundreds are combined they form a unit of one thousand. For example, the answer to 2610 + 4390 is 7 thousands since 610 and 390 combine to form another thousand. Similarly, when a unit of one thousand is ‘decomposed’ into ten hundreds, the number looks different but still represents the same quantity. For example, 4200 can be viewed as 4 thousands, and 2 hundreds, or 3 thousands and 12 hundreds, or 2 thousands and 22 hundreds, etc. Decomposing is used in subtraction problems such as 7200 – 4800 = □ where it is helpful to view 7200 as 6 thousands and 12 hundreds.

At Level 3 students need to develop a multiplicative view of place value that includes understanding the relative size of quantities represented by different numbers. A nested view of 230 as 23 tens allows multiplicative connection between 23 and 230. 230 is ten times larger than 23, and 23 is ten times smaller than 230. Such knowledge can be expressed with equations, 23 x 10 = 230, 10 x 23 = 230, 230 ÷ 10 = 23. Multiplication and division basic facts can be leveraged for harder calculations, 4 x 3 = 12 so 4 x 30 = 120 (ten times more). 30 x 4 = 120 as well. 12 ÷ 3 = 4 so 120 ÷ 30 = 4.

Links to the Number FrameworkThis unit is linked to stages 6 and 7 of the Number Framework.

The learning activities in this unit can be differentiated by varying the scaffolding provided or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. Ways to support students include:

This unit is focussed on the place value structure of whole numbers and as such is not set in a real world context. Learning to read and write numbers in Māori or other Pacific languages will support students’ developing understandings, because number names are derived from their place value structure in these languages.

The activities in this unit could be taught in succession over a number of days to provide a concentrated focus on building place value knowledge. Alternatively, selected activities could be used to support place value understanding while students are working on solving number problems.

Activity 1(1065).

Have them discuss the zero in the number. Then ask them to remove or ‘zap’ another digit in their number and record the value of the digit they are removing. For example: the 6 in the number 1065 is removed. In doing so 60 is being removed or subtracted from the number. This is clearly shown by using arrow cards.

Ask what they noticed when the 6 (60) was removed. (A zero appeared in its place).

Discuss zero as a placeholder. Zero adds no value to the number, but has the important job of holding a place. Have the students make another of the 4-digit numbers and repeat this.

Activity 2Have them play

Fish for 1000(Copymaster 2) in pairs or groups of four to consolidate their understanding of the composition of 4-digit numbers.Students can model the numbers with materials as needed, to support their understanding.

Activity 3words and numerals. For example: ten groups of one hundred, 1000, 10 x 100, one hundred groups of ten, 1000, 100 x 10.Point out that when we write the number 1000 there is no space after the 1 and before the three zeros, whilst with 10 000 a space is used to separate the ten from the three zeros. Highlight the fact that 10 thousand is ten times bigger, by highlighting one row (1000) and then counting the rows.

Ask the students to explain why.

Activity 4numberof the dots that is important, not the size of the dots.Activity 5Activity 6Introduce the game

11,111(Copymaster 4) and support students to play this game in groups of 2 to 4 students.Dear parents and whānau,

In class we have been learning to understand very big numbers, their value and how to read them correctly. It is important for students to be able to read numbers up to a million and beyond, and understand their structure. It would be helpful if your child could practice this at home by browsing real estate or car sales online and reading prices. Challenge them to find the highest and lowest prices advertised.

## Street Maps

In this unit students use street maps as the context to learn about coordinates and grid references, and for giving and following instructions involving directions and distances.

In this unit students are introduced to two ways in which location is represented in real life situations, grid references and coordinates. In both cases the representation is an ordered pair, e.g. F4 as a grid reference, and (6,4) as a pair of coordinates. The convention is to state the horizontal reference first, and the vertical reference second.

In the grid system (left image) the location of the person is an area. Grid references are frequently used on maps. In the coordinate system (right image) the location of the person is a specific point at the intersection of the lines. The coordinate (6,4) can also be interpreted as a vector from the origin (0, 0). The vector is the combined effect of a shift of 6 units in the horizontal direction, and 4 units in the vertical direction.

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

Tasks can be varied in many ways including:

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Using locations that are familiar and of high interest to students supports motivation. The advantage of familiar locations is that students can use their personal knowledge to validate their work. Students often view non-familiar locations through the media. For example, Great Race events, are a hobby for some people, and are readily available on television. Students will enjoy a Great Race though and unfamiliar part of Aotearoa, or through a foreign country. Choose locations that students show interest in, and/or hold particular cultural significance, e.g. Taj Mahal in India.

Session 1Where is the school on the mapWhere is the local park/sports ground.Where is a local shop (dairy, petrol station, bakery, church etc)Where is your house?How would you tell someone else where The Mole and Chicken Restaurant is located?(B3)Where is the Taupo Hospital?(E2)What landmark is in the grid G5(Parts and Service)?Paper maps are not used much these days. Why would that be?Mobile phone technology has made maps redundant and almost removed the need for navigational tools as well.

Can you imagine a place where a map might still be useful?## Session 2

What instructions might we give them?Instructions must include turns (right and left), distances and important things to look for (Street names, buildings, parks, etc.)

How can we find distances on the maps?Students may know about the scale.

This distance on the strip equals 200 metres in real space. Let’s tell our friend how far they need to walk along (Street name). Align the strip with the street beginning a the place where the friend starts walking. For example, “They will need to walk about 300 metres up Taharepa Road.”

## Session 3

What differences can you see between our grid map and this map?Do students notice that the numbers are located on the lines rather than between those lines?

If you went to (5, 2) where would you be?hat is at (6, 2)?(school)What are the coordinates for the Fountain?...the Fire Station?...Ferris Wheel?Put ten new landmarks on the map. Label each landmark, e.g. Hospital, MacDonald’s Restaurant, Skate world.

Write down a set to ten coordinates for the landmarks in the order you want another team to visit them. Try to make the trip an interesting shape.

## Session 4

Why might putting a coordinate system over the Earth’s surface be useful?Students might think of situations where giving a precise location is important, e.g. Flights to a Pacific Island, searching for lost trampers, tracking ships, etc.

Ask:

Do you know where these towns are in Aotearoa?Students might suggest that they are coordinates.

Locate the towns using the coordinates

Use the decimals to get improved precision.

Session 5In this session students create a map that will be of future use to them. They screen grab the map from Google Maps and impose their own coordinate or grid system on it.

Using their own map students might write instructions using cardinal compass directions (N, NE, E, SE, S, SW, W, NW) and orientation instructions. In pairs the students follow each other’s instructions. The activity can be made more challenging by asking students to include an appropriate scale on the map.

Dear family and whānau,

This week we have been learning about using maps with grid references. The game Battleship is a great way to practice these mapping skills. It would be appreciated if you could take some time to play it with your child:

## Figure it Out

## Equality with multiplication and division

This unit of work investigates how relational thinking is applied to multiplication and division.

The equal sign signifies a balance relationship between the numbers and operations on either side.

If the numbers on one side of the equal sign are changed, then relational changes must be made to the other side to maintain equality, e.g. seven is added to both sides.

Understanding the relationship between numbers on either side of the equal sign is fundamental in a range of operational strategies, and to solving of equations later in algebra.

Two progressions are implicated in students’ development to think in relational ways with multiplication and division equations.

Variables Progression

Multiplicative Progression

This unit aims at phases 2-4 of the ‘Variables’ progression and 3-5 of the ‘Multiplicative’ progression.

The learning opportunities in this unit can be differentiated by providing or removing support to students, or by varying the task requirements. The descriptions of each session suggest ways to scaffold students’ thinking and ways to extend the thinking of high achieving students.

Students who find it difficult to understand the properties of multiplication in symbolic form need support with materials so they can appreciate the transformations on the quantities involved. Predictive questioning helps students to think abstractly about concepts rather than rely on physical representations. Symbols are extremely important in connecting different situations, and some students recognise patterns in equations more readily than in other representations.

Able students should be encouraged to generalise the properties of multiplication and how those properties transfer to division, the inverse operation. Their focus should be on identifying and describing operations that can be performed on both sides of an equation that leave the equality invariant.

## Session 1

This session explores how relational thinking applies to multiplication, with a focus on the commutative and distributive properties.

Can students explain which array models 3 x 5 and why? (3 x 5 represents either three rows of five or five rows of three so either answer is justifiable)

Some students may recognise that one array is a quarter turn of the other.

## Session 2

8 x 5 = 4 x 1012 x 4 = 6 x 814 x 3 = 6 x 6

4 x 15 = 2 x 3018 x 50 = 8 x 100

6 x 7 = 2 x 2116 x ? = 8 x 10

12 x 10 = ? x 5

? x 3 = 9 x 6

8 x ? = 4 x 12

24 x 50 = ? x 100

4 x 9 = ? x 3

4 x ♥ = 8 x ♦

♦ x 9 = ♥ x 3

12 x ♥ = ♦ x 6

♥ x 25 = ♦ x 100

## Session 3

÷8 as “the number of rows of eight in 24” and 24÷3 as “the number of columns of three in 24”. Note that division is interpreted as measurement, that is, “How many sets of x go into z?”÷6, since both factors are the same.18 ÷ 3 = ?

36 ÷ ? = 9

? ÷ 5 = 8

28 ÷ ? = 4

48 ÷ 6 = ?

28 ÷ ? = 4

35 ÷ ? = 7

? ÷ 3 = 21

Discuss which facts you look for to solve ?

÷3 = 21.Do your students look in the three times tables to look for a product of 21?

Construct arrays where necessary.

## Session 4

In this final session students work on the relationship between the distributive property of multiplication and strategies for solving division problems.

÷3 = ?What multiplication equation could we write to solve this problem?(? x 3 = 42)Let’s express that in English, both equations ask, “How many threes are in 42?”Where is the answer to both equations?Do your students recognise that 10 + 4 = 14 is the number of fours found in 42?

39

÷3 = ?108

÷9 = ?56 ÷ 4 = ?

75 ÷ 5 = ?

84 ÷ 6 = ?

91 ÷ 7 = ?

Encourage students to record their strategies using multiplication or division equations.

÷4 = 18 then what is 72÷8? There are half as many eights as fours in 72.Dear parents and whānau,

This week in maths we have explore multiplication and division equations. We learn that an equation is a statement of sameness or balance. For example, 6 x 4 = 4 x 6 states than 6 x 4 and 4 x 6 have the same value. Students learn to recognise what can be done to both expressions on either side of the equals sign, yet maintain the equality. For example, 6 x 4 = 6 x 4 can become 6 x 4 = 3 x 8 by doubling and halving the factors on the right. It is also important for students to connect multiplication and division problems. 24

÷ 6 = ?and ? x 6 = 24 both ask “How many sixes are in 24?”## Carrots

In this unit we investigate the amount of water contained in a carrot and we use a time series graph to plot the "weight" of the carrot as the water in the carrot evaporates. The students also pose their own carrot investigative questions.

The key idea of statistical investigations at level 3 is telling the

class storywith supporting evidence. Students are building on the ideas from level two and their understanding of different aspects of the PPDAC (Problem, Plan, Data, Analysis, Conclusion) cycle – see Planning a statistical investigation – level 3 for a full description of all the phases of the PPDAC cycle. Key transitions at this level include posing summary and time series investigative questions and collecting and displaying multivariate and time series data.Summary or time series investigative questions will be posed and explored. Summary investigative questions need to be about the group of interest and have an aggregate focus. For example,

How many seeds are in a carrot packet?What do carrots weigh? Time series investigative questions need to define both the variable and the time period for the investigation. For example,How does the weight of the carrot change over the course of the week?(variable – weight of the carrot; time period – one week).Time series graphsTime series graphs are used to display numerical data over time. The numerical variable of interest is on the vertical axis and the time is on the horizontal axis. For example, the distance travelled each day by an elephant seal from May 2005 to January 2006 is shown in the time series graph below.

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:

The context for this unit can be adapted to suit the interests and experiences of your students. The statistical enquiry process can be applied to many topics and selecting ones that are of interest to your students should always be a priority.

Getting Startedproblem):How much of a carrot is water?How can we find out?(plan)Again get the students to share their ideas.

Weight of carrot at start: _____________ Weight of container: _________

Day

Time

Weight container plus carrot

Weight carrot only

1

Initial

2

Morning

2

Afternoon

3

Morning

3

Afternoon

4

Morning

4

Afternoon

5

Morning

5

Afternoon

Final weight of carrot once dehydrated: __________

data), grate it and put it into the container.data). If this has not happened by the end of the week, resume recording the weight the next week. We still get to find the final weight after the carrot has dried completely to answer our investigative question and we will still have the weight of the carrot over the week to make a time series display in the final session.What are the initial weights of the carrots we have?Data can be collected directly onto a class dot plot on the board or a chart. Ask students to describe the graph by stating what they notice. Include the variable, values, and units in descriptive statements. For example, they might notice that most of the carrots we are using weigh between 80g and 150g. The lightest carrot is 65g.ExploringOver the next three days the students in small groups plan and conduct their own carrot investigation.

ReflectingOn the final day we look at look at and interpret the findings of our first investigation –

How much of a carrot is water. We also share our small group investigations with the class.How much of a carrot is water?analysis).analysis). (There are a few considerations here: has the carrot completely dried? If not, then this reflection session might need to happen in a few days; if they are all completely dried then the final measurement might have been the morning of the 5th day, so a further afternoon reading won’t be needed, or if they have plateaued earlier i.e. fully dried then further recording is not necessary). Students should use a time series graph for this.communicating findings).Small group investigationsDear parents and whānau,

Do you know how much water there is in a carrot?

This week we have been exploring the answer to this and a number of other carrot problems. Ask your child to share their findings to this question plus the other carrot problem they explored.

## Figure it out links

A link from the Figure It Out series which you may find useful is:

## Fridge Pickers

In this unit, using our fridge as the context, we collect data and present these as dot plots and bar graphs. We start to learn about using the computer to display our data.

Planning an investigation at this level is more complex than at Level 2. Students can conduct investigations using the statistical enquiry (PPDAC) cycle. The PPDAC cycle stands for problem, plan, data, analysis, and conclusion. Here the students can begin to talk about situations they have experienced, pose investigative questions, and produce a plan for a statistical investigation. Students may be capable now of incorporating a computer into their work. In this unit the students are introduced to dot plots and bar graphs. Depending on the nature of the investigative questions they are posing they will find out that they need to collect and organise the data in different ways. When considering the data, they now start to see and talk about distinctive features of their displays such as the groupings and modes. Planning a statistical investigation – level 3 provides a full description of all the phases of the statistical enquiry cycle.

Dot plotsDot plots are used to display the distribution of a numerical variable in which each dot represents a value of the variable. If a value occurs more than once, the dots are placed one above the other so that the height of the column of dots represents the frequency for that value. Sometimes the dot plot is drawn using crosses instead of dots. Dot plots can be used for categorical data as well.

Bar graphsIn a bar graph equal-width rectangles (bars) represent each category or value for the variable. The height of these bars tells how many of that object there are. The bars can be vertical, as shown in the example, or horizontal.

The example above shows the types of shoes worn in the class on a particular day. There are three types of shoes: jandals, sneakers, and boots. The height of the corresponding bars shows that there are six lots of jandals, 15 lots of sneakers and three lots of boots. It should be noted that the numbers label the points on the vertical axis, not the spaces between them. Notice too, in a convention used for discrete data (category and whole number data), there are gaps between the bars.

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:

The context for this unit can be adapted to suit the interests and experiences of your students. The statistical enquiry process can be applied to many topics and selecting ones that are of interest to your students should always be a priority.

Session 1We begin the week collecting fridge data about our class. We use this information to start to explore new types of data displays.

What do you think a fridge picker is?Collect ideas from the class and put on the board or a chart.

What types of foods do the students in our class like to pick from the fridge?This is the investigative question (ourproblem).planningfor the investigation we need to establish how we will collect the data that we need to answer the investigation question. We will use the following survey question – but we need to do some work with it so that we have only a few categories, it is possible that every student in the class might have a different choice –What type of food do you like picking from your fridge?Start by getting the students to give responses to the question:What food do you like picking from your fridge?What type of food do you like picking from your fridge – choose from: OPTION 1, OPTION 2, OPTION 3, OPTION 4, OPTION 5, OTHER please specify _________Collect the data.analysis). If possible, take a photo of the display, upload into a google doc or similar and capture student statements about the display.What statements can you make about us as fridge pickers?(analysis)Does the display show us anything different?The distribution of position in the family.What do you notice about this graph? (Take a picture of the new display and add to the document). E.g.More people are the youngest in their family in our class than are the oldest in our class.How is this display different to the one we had when we had sorted by type of food? Does if give us any different information?Communicate findings (conclusion). Refer to the investigative question at the start:What types of foods do the students in our class like to pick from the fridge?and write and answer based on the findings from the explorations above.Preparation for data collection for a home-based activityHow many people in your family do you think are fridge pickers?What time of the day do you think people fridge pick?

How could we find out?

predictionsabout the information they will gather.We think that:

Our fridges will be opened the most before teatime.That no one will open the fridge between midnight and 6 am.

Sessions 2-4Over the next three days the students gather information around the theme of fridges or food. They will collect and record the data in a spreadsheet or data table in a software package that allows them to draw statistical displays. In this unit we use CODAP, but other packages can do similar things. Look for software tools that allow them to have the raw data (i.e. one row containing the responses for each person or object they collect data on) rather than needing summarised data. Using the statistical graphing tool students will display the information using dot plots and bar graphs, and make statements about what the data shows. If you do not have access to computers then provide students with blank templates to make their dot plots (Copymaster 2) or bar graphs (Copymaster 3).

PROBLEM: Generating ideas for statistical investigation and developing investigative questions

Prompts to help with posing investigative questions include:

variableand thegroupand the remaining criteria:PLAN: Planning to collect data to answer our investigative questions

What are the types of fridges we have?they will need to get information to answer this question. Survey questions are how they get the information. In thinking about their investigative question, the students decide they want to find out about the make and the style of the fridge. They pose two survey questions: (1)What is the make of your fridge?and (2)What type of fridge do you have?In discussing with another pair at their table they wondered about what would happen if people had two fridges because one of the students at the table said they had a fridge in the kitchen and one in the garage that was used for drinks and extra food. They decided to add to their survey questions –if you have more than one, pick the one that is your main fridge.For example: Make of fridge question – using the internet they found 15 different makes of fridges so they picked six that they knew and added an “other” category.Categories: Fisher and Paykel, Haier, LG, Samsung, Simpson, Westinghouse, Other

Type of fridge question – using the internet they found the following categories for refrigerators: All refrigerator, Top (freezer) mount refrigerator, Bottom (freezer) mount refrigerator, single-door refrigerator with freezer capacity, Side by side refrigerator, French door model refrigerator, other.Your name

Q1.

What is the make of your fridge? If you have more than one, pick the one that is your main fridge. Choose from:Fisher and PaykelHaierLGSamsungSimpsonWestinghouseOther – please specifyQ2.

What type of fridge do you have? If you have more than one, pick the one that is your main fridge. Choose from:All refrigeratorTop (freezer) mount refrigeratorBottom (freezer) mount refrigeratorSingle-door refrigerator with freezer capacitySide by side refrigeratorFrench door model refrigeratorOther – please specifyRead the directions.Are they clear to you?

Can you tell what you are supposed to do?

What do you think they need to change to make the directions clearer?

DATA: Collecting and organising data

Your name

Q1.

What is the make of your fridge? If you have more than one, pick the one that is your main fridge. Choose from:Fisher and PaykelHaierLGSamsungSimpsonWestinghouseOther – please specifyQ2.

What type of fridge do you have? If you have more than one, pick the one that is your main fridge. Choose from:All refrigeratorTop (freezer) mount refrigeratorBottom (freezer) mount refrigeratorSingle-door refrigerator with freezer capacitySide by side refrigeratorFrench door model refrigeratorOther – please specifyMelino

Simpson

Bottom mount

Ngaire

Fisher and Paykel

Bottom mount

Wiremu

Fisher and Paykel

Bottom mount

Tui

LG

French door model

Anna

Haier

Side by side

Jimmy

Other – Daewoo

Top mount

Jackson

Fisher and Paykel

Side by side

ANALYSIS: Making and describing displays

An introduction to using CODAP as a tool for statistical investigations is explained in full in Planning a statistical investigation – level 3 (session 5).

As mentioned earlier if you do not have access to computers then provide students with blank templates to make their dot plots (Copymaster 2) or bar graphs (Copymaster 3).

Possible examples of the types of information students can get on CODAP is below. In these examples the investigative question

What are the types of fridges we have?is explored by graphing the two variables separately and then combined.These first two graphs show the make and then the type of fridge. Because we are using technology, we can also quickly explore the combination of the two variables.

Once students have generated their graphs, they can start to describe the displays.

To describe the display, encourage students to write

“I notice…”statements about their displays. If students are not sure what to notice the teacher can prompt further statements by asking questions such as:Check the “I notice…” statements for the variable and reference to the group. For example: “

I notice that the most common make of fridge for our class in Fisher and Paykel.”This statement includes the variable (make of fridge) and the group (our class). Support students to write statements that include the variable and the group.CONCLUSION: Answering the investigative question

Tell some of the things you learned from the investigations. What are the preferences of students in our class?What was the most popular choice in your survey?

How can you tell?

How many students made that choice?

Which display is most effective?

Did you have any unexpected results?

Do these displays look alike?Did the two surveys have the same choices for you to make?

What differences are there between the investigations?

Session 5In today’s session we use the data we have gathered at home to examine our initial predictions about fridge pickers.

What can you say about your chart?Which times of the day are the busiest for fridge pickers? Why?

Which times of the day are the quietest? Why?

Who opens the fridge the most often in your home?

Display your data using a dot plotLabel your dot plotGive your poster a titleMake three statements telling things you have learned about the fridge pickers in your family.Dear parents and whānau,

This week at school we are carrying out investigations and making displays of data using dot plots and bar graphs.

For one day at home this week we would like your family to keep track of how often your fridge gets opened. To do this get people to put their initial above the nearest hour each time that they open the fridge. We will be looking at these dot plots at school on ___________.

Provide Copymaster 1.

## Figure it Out Links

A link from the Figure It Out series which you may find useful is: