Numerals and expressions

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Purpose

The purpose of this unit is to develop students’ understanding of numerals as representing a number of items, and their understanding of the symbols for addition and subtraction as representing joining and separating sets of items.

Achievement Objectives
NA1-4: Communicate and explain counting, grouping, and equal-sharing strategies, using words, numbers, and pictures.
Specific Learning Outcomes
  • Understand that an amount or number of items can be represented with a single unique symbol.
  • To correctly write numerals.
  • Understand that written and oral words can also be represented with numeral symbols.
  • Recognise numbers within story contexts.
  • Understand and use the addition and subtraction symbols.
  • Recognise and write addition and subtraction expressions from story contexts.
Description of Mathematics

This sequence of lessons lays a fundamental and important foundation for students to recognise, read and write symbols to record and communicate mathematical ideas. As the symbols become well understood, they also become tools for thinking. This process begins with the introduction of numerals as symbols that represent amounts or numbers of objects. Students hear and see words that are associated with the amounts, and so need to come to understand that a single symbol is representative of all forms of that number, written and spoken, and that the same symbol can represent sets of different items.

Being able to quantify and record amounts is just the beginning. We work on and with numbers. We say we operate on them, and these operations change them. As the symbols that represent the number operations of addition and subtraction are introduced, the students should ‘operate’ on items in real contexts. The language associated with addition and subtraction can be confusing. Students do not always connect "addition", "adding", "and", "plus", or "subtraction", "minus", "takeaway", "less". As many adults use this language interchangeably, students must be supported to connect these operation words with the symbol that represents them.

An expression in mathematics is a combination of symbols (e.g. 4 + 5). An equation is a statement asserting the equality of two expressions (e.g. 4 + 5 = 9). The focus in this unit of work is to have students record expressions, using symbols correctly and with confidence.

In this unit of work, subitising is given an emphasis. The early numeracy stages are defined by a student’s ability to count items, but the ability to subitise or partition an instantly recognised small group of objects into its parts is also important.

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:

  • extending the numbers and numerals in sessions 1 and 2 to nine
  • encouraging students to image the actions on physical materials, by masking or asking anticipation questions, such as “How many ruru will be in the tree then?”
  • providing additional support in session 2, activity 2 by giving students numerals to trace around as they hear them in the story
  • using a calculator to 'record' the operation.

Many of the activities in this unit suggest ways to adapt them to engage with the interests and experiences of your students. Other adaptations include:

  • using te reo Māori words for one (tahi) to ten (tekau) throughout all activities in this unit
  • replacing “Ten in the bed’ with another counting book such as “One Kooky Kereru” by Jahri Jah Jah, or a story that is familiar to your students
  • using objects that are of significance to your students, such as seashells, pictures of their favourite native animals, or items of sport equipment.
Required Resource Materials
Activity

This unit includes several activities for each session. Choose 2-3 activities for a single session or spread each session over more than one day.

Session 1

SLOs:

  • To understand that an amount or number of items can be represented with a single unique symbol.
  • To correctly write numerals to five.
  • To understand that written words and oral words can be represented with a single symbol.
  • To recognise and match words and symbols with the amounts they represent.

Activity 1

Play Number Eye Spy.
(Purpose: To identify numbers of items in groups up to five.)
Explain that in the classroom, some things we see might be in groups of two, three, four or five. Give an example, such as a group of three computers at the back of the classroom. Explain that you could give a clue about the items by saying, “I spy with my little eye, three of something.” You might give directional clues as well to develop the vocabulary of movement and position, e.g., “The items are at the front of the room.” Numbers spoken in te reo Māori can be interchanged with English numbers.

Pose the problem of finding matching collections, by changing the number in the group from three. Have students look about the classroom to identify the group of items you have seen. The person who guesses correctly has the next turn. Take digital photographs so the class can make a display of pictures that match each number from one to five, or further if students show proficiency. Use items that have significance to the students, such as their artworks, cultural artifacts, or toys/mascots. Students might draw collections of objects for a given numeral, such as the number of siblings they have in their whānau or their pets at home.

Activity 2

  1. Play Spot the Spots, using the red and green 1 to 5 dot cards from Copymaster 1a, or the 1 to 5 koru cards from Copymaster 1b.
    (Purpose: To subitise groups of up to five dots.)
    Shuffle the cards. Hold them up one at a time, so that the students can all see them. As each card is held up, have the students show how many dots they see, by holding up the same number of fingers. Students might be asked to represent the number in different ways, such as say the number in words, write the numeral in the air if they know it, or make a set of counters that match the number and layout.
    Image of two cards - one with 3 dots and one with 5 dots.
  2. Shuffle the cards and distribute (at least) one to each student. Going around the mat circle, ask the students to take turns to hold up their card for the others to see. The other students who have a card with the same number of dots hold up their cards too.
  3. Distribute numeral cards to 5, or a number fan to each student. Repeat Step 2 above, but this time the students show the numeral (card) that matches each student’s dot card as it is shown.
  4. The cards can also be used to develop subitising (instant recognition). To encourage imaging, make the card visible for a short time, and ask the student to name the number. 

Activity 3

  1. On the class chart revise together how to correctly write numerals 1 to 5.
    Explain that they will be writing numerals. Discuss what numerals are used for in daily life. Students might suggest things like the number of people who live in a whare, the number of kaimoana on a plate, the number of chicks in a nest, etc.
  2. As numerals are formed on the chart, have students practice forming them in the air, and on the mat in front of them. Have them feel and describe that correct form.
    Emphasise the correct directionality when forming 2, 3 and 5 in particular.
    Model forming each numeral with your back to the students and write the numerals large in the air. Ask students to copy your actions. Adjust your writing hand for students who are left-handed.
  3. Have students return their dot cards from Step 2 by, one at a time, coming to the chart, showing their dot card, saying the number and forming the numeral. Watch for correct numeral formation. Students are likely to need plenty of practice. Students might enjoy writing on small whiteboards or blackboards.

Activity 4

  1. Write numerals 1 to 5 on the class chart. Ask the students what you have written (numerals). Explain that there are words for these numerals that can be written and spoken. Show word cards, one to five, from page 1 of Copymaster 2. Te reo Māori versions are included.
    Hold up the word cards in order and read them aloud together several times. Show them out of order, having the students quickly reading them aloud.
  2. (Using Blue Tac or similar), with assistance from the students, arrange the word cards correctly beside the numerals on the chart. Together check that they match.

Activity 5

Have students work in pairs, or groups of three, to play One, Two, Three, Snap.
(Purpose: To correctly match words, numerals and images of numbers 1 to 5)
Make available to each group, word cards to five (Copymaster 2), numeral cards to 5 (Copymaster 4) and dot or koru images images to 5 (Copymaster 1a or Copymaster 1b). Alternatively, the picture card images from Copymaster 3 can be used instead of the dots. 

Each pile is shuffled and placed face down in front of the group.
Explain that students in the group each take turns to take one card from each pile, placing them face up in front of themselves as they do so. If all cards show the same amount (symbol, word and image) they say “Snap!” and keep the set.
This shows “four” as a word, numeral, and image.
If not, the cards are returned to the bottom of each pile.
The winner is the person with the most sets of three at the end of the game.

Activity 6

Conclude the session with a game of Get Together.
(Purpose: To form groups of up to five in response to hearing a number word, a written number word, or to seeing a numeral or a number word.)
Choose a piece of fast-paced music that your students enjoy. 
Ask your students to stand. Explain that they are to move about the room in time to music. When the music stops, the teacher will either say a number or show numeral or word cards up to five. Students are to look at the teacher, listen for a number or look for a numeral and, as quickly as possible, make a group of that number, and sit down when the group is formed.

Session 2

SLOs:

  • To recognise and match sets of items with written and spoken words and their symbols.
  • To correctly write numerals to five.
  • To recognise numbers within story contexts.

Activity 1

Ensure that numerals 1 to 5 can be seen by the students. Read them together. Begin by playing Spot the Spots from Session 1, Activity 2. Instead of holding up the same number of fingers, have students write numerals in the air and on the mat with their finger.

Activity 2
(Purpose: To accurately write numerals 1 to 5 in response to hearing number words within a story context.)

  1. Make paper and pencils available to the students. Explain that you are going to read them a story (Copymaster 5) and they are to listen very carefully and write down any numbers that they hear in the story. They should write these numerals in order across the page. Alternatively, you could tell them a made up story that includes members of the class.
  2. Introduce the task by reading the first two sentences of the story, show the dog pictures, explain and model what to do.
  3. Read the story, having them complete the task.
  4. Read the story again together, having students actively identify the number words as they are read, and writing the numerals on the class chart. Emphasise correct formation of the numerals.
  5. Have students check each other’s recording.
  6. Ask if students know of other stories with numbers and record their ideas. For example, The Three Little Pigs, Pukeko counts to ten, Counting for kiwi babies, or The great kiwi 1, 2, 3, book. Suggest that students listen and for numbers spoken or seen during the day.

Activity 3

  1. Ensure that students can see number words and numerals.
    Make a think board template (Copymaster 6) and pencil available to each student. Have the students write their favourite number (between 1 and 5) in the centre of their think board. Have them draw a picture, write their own number “story” that includes their favourite number, show any numbers by drawing dots or pictures of equipment, and write any number words, encouraging English and Te Reo.
    You may need to support emerging writers by recording part or all of their story for them, and use digital platforms for students who find handwriting difficult.
  2. Once completed, have student pairs share their thinkboards.

Activity 4

Make the cards from One, Two, Three, Snap (See Session 1, Activity 5) available to the students so they can play a Memory game as they finish their thinkboard. To play memory they should turn all the cards from three sets face down, mix them up and take turns to find matching trios.

Activity 5

Conclude the session with class sharing of thinkboards. Students could take these home to share their learning with their whānau.

Session 3

SLOs:

  • To understand and use the addition symbol.
  • To recognise and use the written and spoken words for addition, with the addition symbol.

Activity 1

  1. Ask the students which numerals they have been learning to write. Extend students to write numerals for 6 – 10. Record these numerals on the class chart. Have students take turns to come and write other numerals that they know on the class chart. As they do so, discuss the numerals, highlighting their correct formation, particularly numerals 6, 7 and 9.
  2. Show a selection of dot cards from 6 to 10 (Copymaster 7a) or koru cards from 6 to 10 (Copymaster 7b), asking students to say what they can see. The focus here is on seeing familiar groups of 1 to 5 dots within the larger group. Some may readily recognise immediately the larger (complete) groups, but this is not the purpose of this task.
  3. Ask students to say the numbers they see in words. Some students may be able to write the words. For example:
    Image of a card with 6 dots arranged in a pattern.“I can see five and one.”
    A card with 5 dots arranged in a diagonal line from the top left to bottom right corner, with one dot is placed in each of the top right and bottom left corners (7 dots total).“I can see five (across) and two (in the corners).”
    Image of a card with 8 dots organised into 2 square groups 4.“I can see four and four.”
  4. Ask, “Is there another way to write this, using numerals?”
    Guide the students to shared recording on the class chart:
    “I can see five and one” → 5 and 1 → 5 plus 1 → 5 + 1
    “I can see five and two” → 5 and 2 → 5 plus 2 → 5 + 2
    “I can see four and four” → 4 and 4 → 4 plus 4 → 4 + 4

Activity 2

Together, make a chart, or class dictionary page, about addition and its symbol, + , asking and recording what the students already know. Use items that students are familiar with to create stories that might be modelled by addition. For example, “Two weka were hunting for worms. Three more weka came along. How many weka were there then?” Record the stories using addition equations, e.g., 2 + 3 = 5 for the weka story.
Create a class book of addition problems that students make up, including a picture and written story. The equation can be written on the back of the page.
Discuss the addition symbol. Included in the ideas recorded, should be:
+ is a symbol or sign
+ shows that we are joining together two amounts or numbers
+ is an addition symbol
when we see the + sign we can read it as “and”, “plus” and “add”
+is a short way of writing “and”, “plus” and “add”.

Activity 3

  1. Together, write a mathematics expression about at least two other dot cards and model several ways of reading what has been written:
    For example: 
    Image of a card with 8 eights divided into a group of 5 and a group of 3.Write 5 + 3 and together read: ‘five plus three’, five and three’, ‘five add three’, ‘five and three more’. 
    You might also highlight that 3 + 5 and the matching statements could also be written.
  2. Make paper and pencils, or, whiteboards and pens, available to the students.
    Distribute at least four dot cards to each child that show numbers of dots greater than 1.
    Have students write, talk or draw about their cards. Use their recording to add to the class chart or create a digital presentation. 

Activity 4

Ask the students work in pairs to play Read and Draw, using the cards from Copymaster 8,
(Purpose: To read a mathematical expression in at least two ways and to respond to a mathematical expression with a drawing.)
Tell the students the purpose of the Read and Draw task. Explain that they take turns to be the Reader and the Quick Draw person.
The Reader’s task is to read the mathematical expression in at least two ways to their partner. Studenrtsshould check their partner’s drawing before showing them the task card on which the expression is written. Their task is to check the accuracy of the drawing.
The Quick Draw person’s task is to listen carefully to the expression that is read, and to draw a diagram of what they hear, using simple shapes. For example: The Reader reads, 4 + 2: “four and two, four plus two,” and the Quick Draw person draws:
Image of 6 triangles arranged in a horizontal line, divided into a group of 4 and a group of 2.
Students should have at least four turns each.

Activity 5

Conclude by playing the Hands Together game using two sets of expression cards from (Copymaster 8).
Each student makes a number of choice on one hand by showing that many fingers. For example:
Image of 2 hands separated by the word “or”. The first hand shows 5 fingers and the second hand shows 3 fingers.
Have them show their ‘number’ to a friend. (This is to avoid students changing the number of fingers when they see the expression.)
The teacher shows an expression, for example 5 + 3. Children who have made these numbers on their fingers must move to pair up, one student showing 5 fingers and the other showing three. The first pair to form and to show 5 + 3 collects a 5 + 3 expression card each. The game begins again. The game finishes when all cards are used up. Students take turns to read aloud to the class the cards they have ‘won’. Each student should read their cards in a different way from the student before them.
Ask students to bring a favourite small soft toy for Session 4. The teacher also needs to bring a small blanket.

Session 4

SLOs:

  • To understand and use the subtraction symbol.
  • To recognise and use subtraction written and spoken words, with the subtraction symbol.

Activity 1

Write this symbol on the class chart: +
Ask students to tell you addition words and give examples of how to use them. For example, “plus”, “I have five fingers on this hand plus five fingers on this hand.” Record these words.

Students might make a given addition expression with materials, like counters, and explain what they have made. Explicitly link the meaning of the numerals, and + and = symbols, with the models that are made.

Activity 2

  1. Together, read the rhyme, “Ten in the Bed.” (Copymaster 9).
    You might choose a different scenario that is appropriate to the interests of your students, such as ruru in the kahikatea tree, or people in the waka.
    Arrange ten of the students’ soft toys in bed, using the blanket.   
  2. Read the rhyme a second time and have the students ‘act out’ one toy falling out each time. Have students take turns to return the toys to the bed, one at a time, and as they do so record addition expressions.
    For example 1 + 1, 2 + 1, 3 + 1, reading these together. As you do so, focus on modeling the correct formation of numerals 6 to 9.
  3. Write 10 and ten (symbol and word) on the class chart and discuss. Read the first verse of the rhyme once more. This time discuss how to use symbols to record what has happened. Write 10 – 1, introducing and recording this as ‘ten takeaway one.’ Continue to read the rhyme, verse by verse, writing each mathematical expression and recording the words each time. Introduce the alternative words for the subtraction symbol as you do so. For example 9 – 1, “nine minus one”; 8 – 1, “eight less one”; 7 – 1, “seven subtract one”.
  4. Ask, “What happened to the number of toys in the bed?” (The number was getting less). Discuss that subtraction symbol, brainstorm and record the student’s ideas of what it is telling us.
    Read the expressions and words again and highlight the different ways we can read the subtraction symbol.

Activity 3

  1. Make available to each student, paper, pencils, five plastic teddies and a tissue.
    Have them put their teddies to bed under the tissue, say the rhyme to themselves and each time a teddy falls out, record the expression, for example, 5 – 1. Those who complete this task quickly can write the addition expressions as they return the teddies to bed, or take more teddies and record expressions for 6 to 10.
  2. Have students pair share, reading aloud their mathematical expressions. As they take turns, encourage the students to use the different language of subtraction, such as “Three take away one equals two.”

Activity 4

Conclude the session with a game of Musical Chairs (or cushions). Use music that is appropriate to the interests of your class, such as hip hop or a current fast-paced popular piece.
Set out the number of chairs for students in the group. Record the number on the chart. Play a favourite piece of music. When it stops all students sit down. Have them stand and ask a student to remove one chair then come to record the expression and read aloud what they have written.
For example 10 – 1, “ten chairs minus one chair.”
Explain that one student will not have a seat this time and that this person will get to write and read the next mathematical expression on the chart.
Continue the game till no chairs remain and subtraction expressions have been written for each action.

Session 5

SLO:

  • To recognise and write addition and subtraction expressions from story contexts.

Activity 1

  1. Using class charts from sessions 3 and 4, review the symbols for addition and subtraction.
  2. Either read the short scenarios from Copymaster 10, exchanging the names of students in the class for those in the scenarios, or create your own. Have the students identify if the story tells of an addition or subtraction ‘event’ and together record these on the class chart as mathematical expressions.

Activity 2

Make pencils, paper, felt pens or crayons available to the students. Have them write at least two of their own scenarios and record the mathematical expressions that represent what is happening. You can motivate them by discussing everyday events in which items are added and subtracted. Contexts might include playing games like tag, using up household items like plates, groups of friends meeting, or birds or other creatures arriving or departing a location, such as penguins in a burrow. Students' illustrations should show what is happening in the scenario and expression.

Activity 3

Conclude the session by having the students pair share, then class share their work. Emphasise the importance of having them read their mathematical expressions in at least two ways.

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