Getting Started
Today we explore upanddown staircases to find the pattern in the number of blocks they are made from.
 Begin the session by telling the students about upanddown staircases:

One block is needed to make a 1step upanddown staircase. It takes one step to get up and one step to get down.


This is a called a 2step staircase as it takes two steps to go up and two steps to go down.

 Together count the steps so that the students understand why it is called a 2step staircase.
How many blocks are in the staircase?
How many blocks do you think would be in a 3step upanddown staircase?
How could you work it out?
 Give the students time to work out the number of blocks. Share the ways that they used to work it out.
 Ask the students to guess how many cubes they think would be needed to make a 5step staircase. Get them to build a 5step staircase. Check their guesses.
 Get the students to build more staircases. As they do, ask them about any patterns they see.
Some may recognise the horizontal layers as being the sequence of odd numbers. Some may see the vertical stacks.
Some may see that square numbers of blocks are always involved. This can be checked by rearranging the blocks to make square numbers (see the diagram below). This is an important discovery. Let them make it. This may require some careful scaffolding on your part.
 Some students may wish to continue to find numbers that make larger upanddown staircases. To keep track of the number of blocks/cubes in each staircase it might be useful to draw the staircases on graph paper. However, if the students prefer to use cubes, then they should record their results in a table.
Exploring
Over the next 23 days the students work in pairs or individually to solve the following problems. Show the students how to use grid paper to continue the patterns. As they complete the problems ask them about any patterns they see and encourage the students to record these observations with the patterns on the graph paper.
Problem 1: Straight up the stairs
How many blocks are in this 4stepup staircase?
How many blocks would there be in a 5stepup staircase?
How many blocks would there be in a 6stepup staircase?
How many blocks in a 10stepup staircase?
How many more blocks will an 11stepup staircase need?
What is the largest up staircase that you can tell us about?
(Note: the numbers of blocks in this pattern are the triangular numbers, see Algebra Information.)
Problem 2: Climbing ladders
How many pieces of wood have we used in this 1rung ladder?
How many pieces of wood have we used in this 2rung ladder?
How many pieces of wood would there be in a 4rung ladder?
How many pieces of wood would there be in a 6rung ladder?
What is the largest ladder that you can tell us about?
How many pieces of wood will you need to add to a 7rung ladder to get an 8rung ladder?
(Note: the number of pieces of wood is three times the number of rungs.)
Problem 3: Small steps
Watch out! You need to take small steps to walk up and down these stairs.
How many blocks are in the 4step staircase?
How many blocks are in the 6step staircase?
What is the largest staircase that you could tell us about?
Does this remind you of something you have done before?
(Note: the count here is the same as that in Problem 1.)
Problem 4: Star patterns



This is a 1star 
This is a 2star 
This is a 3star 
How many blocks are in a 4star?
How many blocks are in a 5star?
What do you notice about the stars?
How many blocks do you need to add to a 7star to make an 8star?
What is the largest star that you could tell us about?
(Note: the pattern here is 1, 5, 9, 13, … At each stage you add on 4 blocks. To make a 100star you need to have 99 lots of 4 plus one block for the centre.)
Problem 5: Lshapes



This is a 1L 
This is a 2L 
This is a 3L 
How many blocks are in a 4L?
How many blocks are in a 5L?
What do you notice about the pattern in the L’s?
What is the largest L that you could tell us about?
(Note: to make a 100L you need 100 + 100 – 1 = 199 blocks.)
Reflecting
In this session we share our solutions to the problems of the previous days. We listen carefully as the patterns are explained. We then make some block patterns of our own which we give to our classmates to continue.
 Begin the session by asking the students to attach their solutions to the problems to a display wall. Give the students time to look at the solutions of other students. Ask for volunteers to share their solutions.
 Give pairs of students a supply of blocks and grid paper and ask them to invent their own block pattern. Tell them to record the first three elements in the pattern on a piece of grid paper.
 Ask the students to swap patterns with another pair. Work together to discover the pattern and then continue it.
 Repeat with another pair’s pattern.
 Leave the patterns on a table for the students to solve in their own time.
Maps
In this unit students are introduced to using maps. They use maps to locate landmarks, identify views from different locations, and give directions using left and right turns and distances.
Maps provide a two dimensional representation of the real world. By looking at a map students should be able to anticipate the landmarks they will see from a given location and in which direction (N, S, E, W) those landmarks will be seen. By using maps of their school or local area students will be able to check their thinking by matching the map with the real world.
Students begin to use the map to help them follow and give directions. They start to use directions involving left and right turns and use landmarks to clarify pathways. Students also begin to use distances in whole numbers of metres.
This unit can be differentiated by varying the scaffolding of the tasks or altering expectations to make the learning opportunities accessible to a range of learners. For example:
Some of the activities in this unit can be adapted to use contexts and materials that are familiar and engaging for students. In particular, the choice of maps to use will depend on the interests of your class. Some students may respond best to maps of familiar areas, while others may be more engaged by an imagined or fantasy context. You could work as a class to create maps of a class favourite story, or the location of a television series or movie.
Session 1
In this session students are introduced to using a map to locate landmarks and identify views from different locations.
Which classroom has the best view of the playground?
What building can you see from the field?
What building can you see out the library windows?
Session 2
In this session students use the school map to describe pathways from locations.
Session 3
In this session students use a local map (or a fictional one) to describe different views they can see from different locations. They use compass directions to give the direction of landmarks from given locations.
How many houses have a direct view of the playground?
What can the children see from the Playcentre?
What can the doctor see out the window?
If you sat in the doctor’s carpark what could you see?
Colour in a house that has a view of the Playcentre, the Dairy, and the Hall?
What building is East of the Café?
What building is North of the Hall?
What building is South of the Chemist?
What direction is the Playcentre from the Church?
What direction is the Playground from the Doctors?
How many houses are South of the Hall?
From which building can you look West to see the Church?
Session 4
In this session students give a set of directions between two locations using distances and quarter turns to the left and right.
Session 5
In this session students learn about pathways and apply this to creating a fire escape plan for their house.
Dear family and whānau,
This week your child has been using maps to describe views and pathways from locations. Your child has started to draw a plan of your house and is finishing it by marking the escape routes out of each room in case of a fire. Please help them to complete the activity.
Figure it out
Some links from the Figure It Out series which you may find useful are:
Staircases
In this unit students look for and describe the patterns they see in different types of staircases.
In much of students’ early pattern work, the numbers involved can be compiled in tables like the one below:
Length of garden
1
2
3
4
5
6
Number of paving stones
8
12
16
20
24
28
Two relationships can be seen:
In practice, recurrence relationships are easier to identify than functional ones.
This unit can be differentiated by varying the scaffolding of the tasks or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example:
The materials used in this unit can be adapted to recognise diversity and student interests to encourage engagement. Instead of creating patterns with plastic classroom blocks or cubes students could be encouraged to make the patterns using environmental materials such as pebbles, shells, or daisies from the school lawn.
Getting Started
Today we explore upanddown staircases to find the pattern in the number of blocks they are made from.
One block is needed to make a 1step upanddown staircase. It takes one step to get up and one step to get down.
How many blocks are in the staircase?
How many blocks do you think would be in a 3step upanddown staircase?
How could you work it out?
Some may recognise the horizontal layers as being the sequence of odd numbers. Some may see the vertical stacks.
Some may see that square numbers of blocks are always involved. This can be checked by rearranging the blocks to make square numbers (see the diagram below). This is an important discovery. Let them make it. This may require some careful scaffolding on your part.
Exploring
Over the next 23 days the students work in pairs or individually to solve the following problems. Show the students how to use grid paper to continue the patterns. As they complete the problems ask them about any patterns they see and encourage the students to record these observations with the patterns on the graph paper.
Problem 1: Straight up the stairs
How many blocks are in this 4stepup staircase?
How many blocks would there be in a 5stepup staircase?
How many blocks would there be in a 6stepup staircase?
How many blocks in a 10stepup staircase?
How many more blocks will an 11stepup staircase need?
What is the largest up staircase that you can tell us about?
(Note: the numbers of blocks in this pattern are the triangular numbers, see Algebra Information.)
Problem 2: Climbing ladders
How many pieces of wood have we used in this 1rung ladder?
How many pieces of wood have we used in this 2rung ladder?
How many pieces of wood would there be in a 4rung ladder?
How many pieces of wood would there be in a 6rung ladder?
What is the largest ladder that you can tell us about?
How many pieces of wood will you need to add to a 7rung ladder to get an 8rung ladder?
(Note: the number of pieces of wood is three times the number of rungs.)
Problem 3: Small steps
Watch out! You need to take small steps to walk up and down these stairs.
How many blocks are in the 4step staircase?
How many blocks are in the 6step staircase?
What is the largest staircase that you could tell us about?
Does this remind you of something you have done before?
(Note: the count here is the same as that in Problem 1.)
Problem 4: Star patterns
How many blocks are in a 4star?
How many blocks are in a 5star?
What do you notice about the stars?
How many blocks do you need to add to a 7star to make an 8star?
What is the largest star that you could tell us about?
(Note: the pattern here is 1, 5, 9, 13, … At each stage you add on 4 blocks. To make a 100star you need to have 99 lots of 4 plus one block for the centre.)
Problem 5: Lshapes
How many blocks are in a 4L?
How many blocks are in a 5L?
What do you notice about the pattern in the L’s?
What is the largest L that you could tell us about?
(Note: to make a 100L you need 100 + 100 – 1 = 199 blocks.)
Reflecting
In this session we share our solutions to the problems of the previous days. We listen carefully as the patterns are explained. We then make some block patterns of our own which we give to our classmates to continue.
Dear parents and whānau,
In maths this week we have explored different block patterns.
The home task this week is for your child to continue this pattern.
Figure it Out Links
A link from the Figure It Out series which you may find useful is:
Data cards: Level 2
This unit introduces the students to a way of looking at information from a group of individuals, i.e. a data set. “Data cards” are used to display information about individuals and by sorting and organising a set of data cards, students can find out things or answer questions about the group.
A “data card” is simply a square piece of paper containing information about an individual person or thing. At this level the data card is divided into three areas with the same category information in the same location on each card. In this unit the terms data and information are used to mean the same thing and are interchanged throughout. Because several pieces of information about individuals are on each data card, different categories can be looked at simply by rearranging the cards.
This unit focuses on sorting and organising data sets, i.e. collections of information from a group of individuals. As the data set is looked at, questions or interesting things arise, which is different from starting with an investigative question then collecting data to answer the investigative question.
Understanding the difference between individual data and group data is central to the unit. The goal is to move students from “that is Jo’s data and that is me” to making statements about the group in general. Increasing students' ability to accurately describe aspects of a data set, including developing statistical vocabulary, is part of the unit. As students become comfortable with making statements and describing data, more precise vocabulary is to be encouraged. The meaning and usage of words like; same, similar, exactly and almost need to be explored during the unit along with the importance of using numerical descriptions, e.g. 2 more than, when describing or comparing data.
Investigative questions
At Level 2 students should be generating broad ideas to investigate and the teacher works with the students to refine their ideas into an investigative question that can be answered with data. Investigative summary questions are about the class or other whole group. The variables are categorical or whole numbers. Investigative questions are the questions we ask of the data.
The investigative question development is led by the teacher, and through questioning of the students identifies the variable of interest and the group the investigative question is about. The teacher still forms the investigative question but with student input.
Data collection or survey questions
Data collection or survey questions are the questions we ask to collect the data to answer the investigative question. For example, if our investigative question was “What ice cream flavours do the students in our class like?” a corresponding data collection or survey question might be “What is your favourite ice cream flavour?”
As with the investigative question, data collection or survey question development is led by the teacher, and through questioning of the students, suitable data collection or survey questions are developed.
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.
Session One
Does anyone in the class fit this data card?
Do you know someone that fits this data card that is not in this class?
How many different people could this data card be correct for?
What could “right handed” mean?
If the data card just said “brown”, instead of “brown eyes” what could it mean?
What would a data card about you look like?
Session Two
Initially encourage the students to look at one category at a time then, encourage students to look for categories within other categories, e.g. What hands do girls write with?
Session Three
What do you think we will find out about our class?
Will it be mainly different or similar to the group looked at in Session Two?
Session Four
Today the students, in pairs, will design and collect their own data using data cards. Each pair of students needs to design two data collection questions to ask other students in the class. The first data collection question will be “Are you a boy or a girl?” with two new data collection questions added.
Sample data collection questions:
Session Five
In pairs the students are to sort and organise their 24 data cards to look for other interesting things about the class and to see if the statements they made about the class were correct.
After a set time each pair reports what they found out about the class. This could be in the form of a written report, a conference with the teacher or a presentation to the class.
Getting partial
In this unit we explore fractions of regions as well as fractions of sets. We look for and develop understanding of the connection between fractions and division.
Fractions are probably the first departure from whole numbers that students will see. This unit introduces a number of important concepts relating to fractions. The first of these is that fractions represent parts of one whole, and can be represented in a variety of ways including regions and sets. This makes them useful in a large variety of situations where whole numbers by themselves are inadequate.
The second useful concept is that a given number can be represented as a fraction in many ways. Knowing that fractions such as ½ can be disguised as 2/4 or 3/6, etc. is important both for recognition purposes and for use in calculations.
Finally, students should know that fractions can be represented both as one whole number divided by another whole number and as points on the number line. Having a knowledge of the different representations of fractions provides connections across mathematics for students and so increases their level of understanding.
In this unit we also introduce the idea of a fraction of 100 and so lay the groundwork for the decimal representation of fractions at Level 3 and percentages at Level 4. These ideas are developed further in the units Getting the Point, Level 3 and Getting Percentible, Level 4. Facility with fractions is also an important precursor for algebra. Algebraic fractions have a wide range of uses. Without a good knowledge of how fractions work, students will be restricted in their work at higher levels of the secondary school when fractions occur in algebraic settings.
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. For example:
The context for this unit can be adapted to recognise diversity and student interests to encourage engagement. For example, students in kapa haka teams could be used as a context for fractions of sets, and eels could be used instead of worms when considering fractions of lengths.
Session 1
Here we look at different representations of 1/2.
Session 2
Here we look at fractions other than 1/2 and consider ways to represent these fractions that involve 100.
Session 3
This session involves fractions in problem situations.
Session 4
Another way to represent numbers is the number line. Here we use the number line to show the relative positions and sizes of fractions.
Pose these problems for students to solve using their strips.
These problems will highlight students’ knowledge of the relative size of fractions. For example, a student might find half of the distance between 0 and 1/5 to see where 1/10 should be or half the distance between 1/2 and 1 to see the location of 3/4 . The problems will also highlight their understanding of the role of the numerator (top number) as the selector of the number of parts and the role of the denominator (bottom number) as nominating how many equal parts the whole is separated into.
Session 5
Here we try to link the concepts of fractions in length and sets by dividing up a big worm.
The worm was 18 cubes long. Each bird got three cubes of worm. How many birds were there?
Dear family and whānau,
This week we have been thinking about fractions. Ask your child do explore some fractions with you. For example: take a paper or a magazine and ask them to find the longest word that they can. How many letters does it have? Now find some words that are half that length or a third of the length or a quarter of that length. Ask your child to record the words they find and the fractions you talk about.
How many pages does the paper or magazine have? What is half that number and a quarter of that number?
We would be glad if the answers could be brought back to class so that we can discuss them.
Figure it Out Links
Some links from the Figure It Out series which you may find useful are:
Scavenger hunt
In this unit students participate in a series of scavenger hunts to develop their own personal benchmarks for measures of 1cm, 10cm, 50cm and one metre. An understanding of the relationship between centimetres and metres is also developed.
Children need to be able to see the need to move from using non standard measures of length to standard measures of length. The motivation for this will arise out of students comparing differences in the length of their hand spans etc. From this the need for standard measurement will become evident.
Students also need to develop personal benchmarks with which to measure various objects in their daily lives. Their personal benchmarks need to gradually relate more to standard measures such as metres, 1/2 metres.
The ultimate aim is for students to be able to choose appropriately from a range of strategies including estimation, knowledge of benchmarks, and knowledge of standards measures in order approach various measuring tasks with confidence and accuracy.
This unit can be differentiated by varying the scaffolding of the tasks or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example:
This unit can be adapted to acknowledge student interests, encouraging engagement. The scavenger hunts could be carried out in different locations, for example, in the classroom, in the school playground, at a local park or beach.
This unit is run as a series of stations over four days with students rotating around the stations in groups. The final session is run as a class activity with all students working on the same task in groups.
The four stations involve the students looking for objects that they estimate to be a certain length. You will need to set appropriate boundaries for their search, e.g. the classroom or the playground.
As students work, the teacher can circulate amongst the groups. Points to reinforce in your discussions with students include
How many 1 cm lengths in a metre?
How many 10 cm lengths in a metre?
Why is 50 cm sometimes called half a metre?
What is another name for a metre?
Station One
Students work in pairs or small groups to find items that they estimate to be 1cm long. They check their estimates by measuring.
Student Instructions (Copymaster One)
Go on a Scavenger Hunt!
Object with estimated length 1cm
Measured length
How accurate were your estimates?
Were your estimates too long or too short?
What would be a good way to try and remember how long 1cm is?
Station Two
Students work in pairs or small groups to find items that they estimate to be 10cm long. They check their estimates by measuring.
Student Instructions (Copmaster Two)
Go on a Scavenger Hunt!
Object with estimated length 10cm
Measured length
How accurate were your estimates?
Were your estimates too long or too short?
What would be a good way to try and remember how long 10cm is?
Station Three
Students work in pairs or small groups to find items that they estimate to be 50cm long. They check their estimates by measuring.
Student Instructions (Copymaster Three)
Go on a Scavenger Hunt!
Object with estimated length 50cm
Measured length
Difference between estimated and measured length
How accurate were your estimates?
Were your estimates too long or too short?
What would be a good way to try and remember how long 50cm is?
Station Four
Students work in pairs or small groups to find items that they estimate to be 1metre long. They check their estimates by measuring.
Student Instructions (Copymaster Four)
Go on a Scavenger Hunt!
Object with estimated length 50cm
Measured length
Difference between estimated and measured length
How accurate were your estimates?
Were your estimates too long or too short?
What would be a good way to try and remember how long 1 metre is?
Reflecting – Class activity
At the conclusion of the session reveal the correct letters for the 1cm, 10cm, 50 cm and 1 metre lengths. Students check their answers and have a chance to measure the strips they chose as required.
Extension
Students who finish the activity early could estimate and measure the lengths of the other paper strips at the stations.
Family and whānau,
This week in maths we are working on estimating lengths of up to a metre. Can you please help your child find an objects at home that they estimate to be 1cm, 10 cm, half a metre, and 1 m long? They can record the names of the object and the estimations in their book. Ask them to choose 1 object to bring to school so we can measure it carefully to check their estimation.