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Equality and Equations

Keywords:
Achievement Objectives:

Achievement Objective: NA1-4: Communicate and explain counting, grouping, and equal-sharing strategies, using words, numbers, and pictures.
AO elaboration and other teaching resources
Achievement Objective: NA2-6: Communicate and interpret simple additive strategies, using words, diagrams (pictures), and symbols.
AO elaboration and other teaching resources

Purpose: 

The purpose of this unit of five lessons is to develop the algebraic understanding that the equals symbol, = , indicates a relationship of equivalence between two amounts.

Specific Learning Outcomes: 
  • Review number expressions involving the operations of addition and subtraction.
  • Make and recognise combined amounts that have the same value.
  • Write statements of equivalence in words.
  • Read and write addition and subtraction equations.
  • Solve addition and subtraction balance problems and explain the solutions, using the language of equivalence.
  • Recognise expressions that are equal in value.
Description of mathematics: 

This sequence of lessons lays a fundamental and important foundation for students to be able to read, write and understand an equation. The most common understanding of students as they come to work with equations, is that the equals sign indicates that the answer is what follows. Students develop early the perception that this symbol, =, is a signal to do something.

The essence of an equation is that it is a statement of a relationship between two amounts. This relationship is a significant one of equivalence. That amounts on either side of the equals sign are equal in value, is an essential understanding that must be in place if students are to experience success (and enjoyment) in algebra in particular, and in mathematics in general, into the future.

Many students in senior primary school and early secondary classrooms, struggle to solve even simple algebra problems because they have a process view of an equation as a procedure to follow to get an answer, rather than a structural or relational view of equivalence. The more established this understanding is, the more difficult it becomes to change the student’s process view, to that of seeing an equation as an object that can be ‘acted upon’.

Students should be immersed in a range of experiences that support them to explore the concept of equivalence and balance. During these experiences, the teacher must carefully choose the language they use and model. As equations are introduced, recorded, read and interpreted, words and phrases such as ‘has the same value as’, ‘is the same as’, ‘is equal to’ and ‘ is equivalent to’, rather than ‘makes’, or ‘gives an answer of ’, become very important. It is interesting to note that the word ‘equals’, on its own, has subtly become more synonymous with ‘makes’ or ‘gives an answer’, rather than giving the message of equivalence that it should. Note that the words ‘addend’ and ‘sum’ are introduced in Number units of work, rather than in this series of lessons.

When posing problems that position the unknown amount at the beginning or in the middle of an equation, we are challenging the student to explore the relationship statement and the operations from a different perspective. This also occurs when students are asked to find ‘different names’ for the same amount. Further, to require students to discern true statements from false, and to justify their choice, pushes them to think more deeply about the nature of the relationship expressed in an equation.

Students should have opportunities to both read and respond to equations, and to record them, having interpreted a number problem expressed in words. In developing the ‘balance’ view of an equation, students will understand the equality relationship expressed in an equation such as 6 = 6, rather than being perplexed by the fact that there is no number problem to ‘answer’. Students will also readily understand relationships expressed in equations such as 4 + 2 = 1 + 5, rather than developing an expectation that a single ‘answer’ will follow the = symbol. Instead of expressing solutions in the arithmetic ‘voice’ of ‘problem, calculation and answer’, it is important in early algebra work, for students to explain their solutions in words that make the equivalence relationship explicit.

The activities suggested in this series of lessons can form the basis of independent practice tasks.

Links to the Number Framework
Counting all (Stages 2 – 3)
Counting on (Stage 4)

Required Resource Materials: 
class set of balance scales
plastic coloured beans
plain envelopes
Jack and the Beanstalk story

plastic containers
plastic pegs
cardboard (labels)
Activity: 

These learning experiences use numbers in the range from 1 to 20, however the numbers in the problems and the learning experiences should be adapted, as appropriate, for the students.

Session 1

SLOs:

  • Review number expressions involving the operations of addition and subtraction.
  • Make and recognise combined amounts that have the same value.
  • Write statements of equivalence in words.
  • Write and read equations, using the language of equivalence.
  • Understand the word ‘equation’

Activity 1

  1. Introduce the story of Jack and the Beanstalk. Ask who has planted or picked beans. Read the story. Explain that when the beanstalk is chopped to the ground, Jack picks handfuls of beans from it, some of which are bright green and others dark green. Unfortunately, they are no longer ‘magic’.
  2. Draw on the class chart, the combinations of beans in Jack’s handfuls. Have students record beside them, in words and number expressions, what they see. For example:
    three and four beans 3 + 4
    two plus five beans 2 + 5
    Pose subtraction scenarios and have students record their number expressions.
    For example:
    “Jack has eight beans and drops four.” 8 - 4
    “Jack has 6 beans and drops 1.” 6 - 1

Activity 2

  1. Make available to the students pencils, envelopes, and sets of two different coloured beans. Have students work in pairs.
    Pose the problem:
    “Jack wants to give away some packets of beans. He decides he’ll put six in each packet. He put some beans of each colour into each packet and writes on the outside of the packet how many there are of each colour."  
    Write 6 on the class chart.
    Demonstrate. For example:
    Put  into one  and write 2 + 4 in pencil on the outside. 
    Tell the students that they should take turns to put the beans into the packets and to write on the outside.
  2. Have students pair share their packets and discuss if they have the same combinations recorded. Have them investigate any anomalies. (They may have put more than six in a packet).
  3. Have student pairs return to the mat with their bean packets, which they place in front of them. On the class chart record:
    6 is the same amount as:
    Have students take turns to record their number expressions beside this.
    6 is the same as: 5 + 1, 4 + 2, 3 + 3, 2 + 4, 1 + 5
    Read these together using the language of, “is the same as.”
    Ask whether it would be fair for Jack to give these to his friends. (Yes, because they would be getting the same amount. They would be getting an equal amount.)

Activity 3

  1. Write the word ‘equal’ on the chart.
    Have students tell you what ‘equal’ means. Brainstorm ideas and record these.
  2. Add to the recording in Activty 2, Step 3.
    6 is the same (amount) as: 5 + 1, 4 + 2, 3 + 3, 2 + 4, 1 + 5
    6 is equal to: 5 + 1, 4 + 2, 3 + 3, 2 + 4, 1 + 5
    Ask if students know how to write “is equal to” using a symbol. Introduce =.
    Model writing 6 = 5 + 1.
  3. Have student pairs share the task of writing the complete equation on each of their packets.
  4. Have students place their “6” packets into a class container to be used in a later session.

Activity 4

  1. Place these containers in front of the students.

    Explain the Jack needs packets with these different amounts. Demonstrate, using a ‘six packet’, that each envelope must have the “number equals story” on it.
  2. Ask what is the correct word for a “number equals story”. Elicit and record the word equation, highlighting that 6 = 5 + 1 (for example) is called an equation because it uses the = sign to show that both amounts are the same. Ask if they can see part of the word ‘equals’ in the word equation.
  3. Have students make up packets, as before, this time choosing 3, 4, 5, or 7 as their total, and recording a full equation on each packet. For example, 3 = 2 + 1, or 4 = 2 + 2.
  4. Students should pair share and check their packets and equations before they are placed in the appropriate containers.

Activity 5

Conclude the session by reviewing =, equals and its meaning and the meaning of the word equation. Have students explain these, and record what they say.

Session 2

SLO:

  • Write and read addition equations, using the language of equivalence.

Before the session, halve the number of packets in each of the containers and prepare 2 sets.

Activity 1

  1. Review the words, equal, equation and the symbol =, recorded on the class chart in Session 1.
    Record a ‘six’ equation and read it in different ways together. For example:
    6 = 5 + 1, “six is equal to five plus one”, “six is the same as five plus one”.
    Highlight the fact that each of the packets in the 6 container have an equal or same amount.
  2. Make available to the students, pencil and paper.
    Have students in pairs choose one of the containers.
    Students begin by taking turns to read aloud to their partner, in the two ways modeled in Step 1 (above), an equation on an envelope selected from the container. They should return these once read.
  3. Explain that in shops, staff do ‘stocktaking’ to check the amount of items they have. Students are to “stock take’ the beans by checking each packet to see that the equation on the outside matches the beans inside.
    They should take two packets at a time, check that they have exactly the same amount and record what they find on their “stocktaking sheet” like this:

    Students with containers 3 and 4 in particular, will accomplish this quickly.

Activity 2

  1. Have each student pair join one other pair in this way: two pairs of seven and three, two pairs of six and four and one pair of five and five.
  2. Refer to Jack and the Beanstalk.
    Place in front of the students the cardboard ‘tickets’ and the plastic pegs.

    Pose the task:
    “Jack is going to have a bean stall. He needs ‘pegged pairs’ with ten beans altogether in each. We are going to help him We need to make labels to show the contents, or what's inside." 
    Elicit from the students that by using one packet from each of their containers, they will have ten beans. If necessary, students can explore this idea and check, using their fingers, showing, for example: 10 = 7 fingers (up) and 3 fingers (down).
  3. Demonstrate that the two packets can be pegged together to make one “pegged pair of ten.” Model on the class chart, how labels should show the content in 3 ways. For example:

    a. Write an equation using the number on each of the containers.
     We say "7 beans plus 3 beans equals 10 beans" and we write 7 + 3 = 10
    b. Write each of the expressions written on each envelope. (The number of each colour in each envelope)
    We say: "This envelope has 5 dark and 2 light and this envelope has 1dark and 2 light. Altogether that equals 10. " We write:  5 + 2 + 1 + 2) = 10
    c. Tip out the beans and write the number of each of the colours.
    We say" There are 6 dark beans and 4 light beans and that is 10 beans altogether." We write: 6 + 4 = 10
  4. Conclude the session by having the students read aloud some of the tickets they have made for their pegged pairs.
    Review the words, equal, equation and the symbol =, recorded on the class chart in Step 1, highlighting the language of ‘is equal to’ and ‘is the same as’ and that all the equations written are different names for ten.

Session 3

SLOs:

  • Solve addition and subtraction balance problems and explain the solutions using the language of equivalence.
  • Read and write addition and subtraction equations.

Activity 1

  1. Introduce balance scales. Brainstorm and record on the class chart, students’ ideas about ‘how balance scales work’, eliciting language of ‘same, level, equal, balance, not tipped.’
  2. Place one envelope pair (10) in one pan and ask what could be placed in the other to achieve balance. (Another pegged envelope pair.)
    Again, record and ‘test’ student ideas, trying different combinations of pegged pairs. For example:
    5 + 5 = 6 + 4
    6 + 4 = 7 + 3
    Ask why the results are recorded using =.
    Elicit reasons such as ,”equals shows that they are the same”, “equals shows that they balance”, “equals shows that both amounts have the same value (10)” , “equals means is the same as”.
  3. Record, 10 = 10 and discuss why this has been written and why is makes sense.

Activity 2

  1. Model 5 + 5 = 6 + 4 using the scales.
    Remove the packet of 4 beans, leaving 6 only on one side. Discuss the tipped scales and how to record the removal of the 4 beans.
    Record suggestions. For example:
    5 + 5 is not the same as 10 – 4
    5 + 5 is not equal to 10 – 4
    10 is not equal to 6
  2. Ask what can be done to restore the balance.
    Accept, ‘put 4 back in again’, but work to elicit, ‘take 4 away from the other side.
    Have a student remove 4 beans from one of the 5 bean envelopes (example above), saying how many are remaining in the envelope (1). Return it to the scales.
    Record suggestions that describe what has happened now the balance is restored. For example:
    5 + 5 - 4 is equal to 10 – 4
    10 - 4 is the same as 10 – 4
    10 – 4 = 10 – 4
    6 = 6
    As equations are recorded, have students explain or demonstrate, using the materials, exactly what is happening. Together reach the conclusion: if you take away the same amount from each ‘side’ or pan, the scales will still balance.
  3. Make available to the students, fresh envelopes (or erasers to clear used envelopes), and pegged bundles of ten from Session 2.
    Have student pairs combine the beans from the pegged pairs into single envelopes of ten beans, writing 10 on each.

    Have students work in pairs with envelopes of ten beans, some spare beans, paper to record equations and a set of balance scales.
    Have students undertake the following tasks.
    A. Student One removes a number of beans from one envelope, unseen by the other student, and returns the envelope to the scales. This student ‘secretly’ records the equation. For example: 10 – 3 = 7.
    Student Two guesses how many were removed, removes this number from the other envelope, ‘secretly’ records the equation, for example 10 – 5 = 5, and returns it to the scales. They look carefully to check to see if the scales balance. If the scales do not balance, Student Two repeats their turn with another amount. When the scales do balance, both students share their final equations and check the amount in each envelope. Both students finally record the balance, for example, 7 = 7.
    The students reverse rolls.
    B. Students make teen numbers and record equations.
    Student One places one ten envelope and a mixture of both colours of beans into one pan to make a number between ten and twenty. The student records the equation: for example, 10 + 2 + 3 = 15.
    Student Two places one ten envelope and a mixture of both colours of beans into the other pan. The two-bean mix must be a different combination, but the total must balance the scales (in this case must equal 15). This student records their equation: for example, 10 + 1 + 4 = 15.
    Both students then record what they can see in both pans.
    10 + 2 + 3 = 10 + 1 + 4
    15 = 15
  4. Conclude this session with some students sharing their equations from tasks A. and B. Record a selection on the class chart and discuss these.
    It is important to highlight the balance nature of the equations. Elicit from the students, ‘an equations is like a thing that balances.’

Session 4

SLOs:

  • Interpret addition and subtraction word problems that involve start unknown, change unknown and result unknown amounts.
  • Write addition and subtraction equations from word problems.

Activity 1

  1. Review conclusions from Session 3, Activity 2, Step 3, referring to the balance scales.
  2. Make available to the students: balance scales, packets of beans, spare beans, and a pencil.
    Explain that Jack, of Jack and the Beanstalk fame, has some problems for the students to solve and that they may want to use the equipment to help them.
    Distribute a copy of Attachment 1 to each student. Read through the problems together.
    Highlight that each student will be writing equations for each problem.
    Students should choose whether to work on the problems alone or with a partner however, each student should complete their own recording sheet.
  3. As students complete the recording task, have them compare and discuss their equations and solutions. They can then write some problems for their partner to solve.

Session 5

SLOs:

  • Identify true (correct) from false (incorrect) equations and justify the choice.
  • Recognise expressions that are equal in value.

Activity 1

  1. Students will play two games in the session. Make available beans and balance scales.
    Introduce the True/False game. (Attachment 2)
    (Purpose: To recognise when amounts are equal or not equal.)
    Model a ‘true’ equation such as 1 + 3 = 2 + 2, highlighting the fact that the amount on both sides are the same or equal to each other. Each expression is equal to 4. Model a ‘false’ equation such as 1 + 3 = 3 + 2, highlighting the fact that both sides are not the same and not equal to each other. 4 is not equal to 5. This is false (not true).
  2. How to play:
    Students play in pairs. They shuffle the playing cards and deal 10 to each player. The remainder of cards is placed in a pile, face down, handy to both players.
    The aim of the game is to be the first person to have an equal number of true and false equations (five of each).
    As each player turns over their cards, they sort them into true and false groups, face up in front of themselves. If they have more of one group than the other, they continue to take cards from the top of the pile, till the number of their true and false cards is equal.
    The first player to have equal numbers of true and false cards calls, “Stop!”
    This caller must explain to their partner, for each of their decisions, how they know they are correct in their true/false decisions. They can use beans to support their explanation.
    The game begins again. The winner is the person who wins more of three games.

Activity 2

Students play Same Name snap, using cards from Attachment 3.
(Purpose: To recognise when amounts are equivalent (or not equivalent) and to give the ‘number name’ for the ‘same name’ expressions.)
How to play:
Student pairs shuffle the cards and deal all cards so each student has an equal number of cards. These are placed in a pile, face down in front of each student. Student One turns over the top card and places it, face up, between both students. Student Two does the same, placing their card on top of their partner’s card. If the two expressions have equal value, either student calls Same Name, states the number that the expression represents, and the correct equation using either ‘is equal to’ or ‘is the same as’. For example:
2 + 3 is placed on top of 4 + 1.
“Same name! Five! Two plus three is equal to four plus one.” or 
“Two plus three is the same as four plus one.”
The caller collects the card pile, records the equation, 5 = 2 + 3 = 4 + 1 on their scoring paper, and the game begins again, with the winner of this round placing the first card.
The student who does not call, can challenge the caller if they believe the “name” is not true for either or both expressions. If they are correct, they collect the pile and record the correct equation. The original caller must erase the incorrect equation.
The game finishes when one student has all the cards, or when one student has recorded ten ‘same name’ equations.

Activity 3

Conclude this session by discussing learning from the games, and reviewing ideas recorded on the class chart over five sessions.

AttachmentSize
EqualityAndEquationscm1.pdf135.49 KB
EqualityAndEquationscm2.pdf24.68 KB
EqualityAndEquationscm3.pdf25.79 KB