Equality

Equality
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Foundational Webinar

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View the Foundational Webinar "Insight into Equality".

Recording

Webinar Slides: click here to open (PDF)

Webinar Guide : click here to open (PDF)

 

 "After watching the Equality webinar, I now realize why my first year Calculus students and my 5 years old son are making the same mistakes related to equality."

- Sean Graves

 

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After watching the archived webinar, please fill out this short survey.

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Key understanding

Quote

The National Council of Teachers of Mathematics states, "In understanding equality, one of the first things students must realize is that equality is a relationship, not an operation" (2000–2007).

http://www.learnalberta.ca/content/mepg7/html/pg7_preservationofequality/step1.html 

What Is Equality?

Equal in math refers to a relationship between objects that can be expressed with a number.  These sets contain the same amount of objects, expressed as numbers (whole or rational) or as units of weight, units of length, units of area, units of volume.


Equal is not just sameness; it is same amountness.

Equality

 

Caution about the “Balance” Metaphor:

Balance, which measures mass, is the metaphor predominantly used for teaching equality.  However, equality is not just about balance. We need to use a variety of models and/or metaphors in order to ensure that other attributes of equality are measurable.  When using the balance metaphor, we want to ensure students understand that we use the term as an adjective to describe the state of equality rather than a verb.

Why Is Equality Important?

MathDetermining equality and inequality is the foundation for everything we do in math.  It is used when working with number lines, comparing, ordering and determining magnitude and developing an understanding of commutative, associative and distributive properties.  Equality is important because kids don’t see that equality has to do with the number of things and not the physical attributes such as size, mass, etc.

According to the Alberta Mathematics Kindergarten to Grade 9 Program of Studies (2014), Change and Relationships are two of the components that define the Nature of Mathematics.

 

CHANGE

It is important for students to understand that mathematics is dynamic and not static. As a result, recognizing change is a key component in understanding and developing mathematics.

 

RELATIONSHIPS

Mathematics is one way to describe interconnectedness in a holistic worldview. Mathematics is used to describe and explain relationships. As part of the study of mathematics, students look for relationships among numbers, sets, shapes, objects and concepts. The search for possible relationships involves collecting and analyzing data and describing relationships visually, symbolically, orally or in written form.

https://education.alberta.ca/media/8775377/k_to_9_math_pos.pdf

 

Mathematics Kindergarten to Grade 9 Program of Studies, Alberta Education, 2014

Why Begin with Equality?

It is important that students first understand relationships between numbers before they are asked to operate with them.  Operations emerge as a thinking and communication tool to help us in our quest to determine and prove equality and inequality.  The equals sign naturally arises through a need for the formal presentation of determining equality.

Teacher Background Knowledge

What do I need to know as a teacher in order to be able to teach the concept(s)?

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Vocabulary

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Program of Study Outcomes

K-9 Program of Studies (PDF)

Below you will find outcomes from k - 6 related to Equality.  There is room for conversation about how/if different outcomes are related to equality….   did all teachers choose the same outcomes?

Kindergarten:

N2

Subitize (recognize at a glance) and name familiar arrangements of 1 to 5 objects or dots

N5

Compare quantities 1 to 10, using one-to-one correspondence

 Grade 1:

N4

Represent and describe numbers to 20, concretely, pictorially and symbolically

N5

Compare sets containing up to 20 elements, using

  • Referents
  • One-to-one correspondence

To solve problems

N7

Demonstrate an understanding of conservation of number

P/R 4

Describe equality as a balance and inequality as an imbalance, concretely and pictorially (0 to 20)

P/R 5

Record equalities, using the equal symbol

Grade 2:

N5

Compare and order numbers up to 100

P/R4

Demonstrate and explain the meaning of equality and inequality, concretely and pictorially

P/R5

Record equalities and inequalities symbolically, using the equal symbol or the not equal symbol

Grade 3:

N3

Compare and order numbers to 1000.

N13

Demonstrate an understanding of fractions by:

  • explaining that a fraction represents a part of a whole
  • describing situations in which fractions are used
  • comparing fractions of the same whole that have like denominators

P/R4

Solve one-step addition and subtraction equations involving a symbol to represent an unknown number.  

Grade 4:

N2

Compare and order numbers to 10 000.

N6

Demonstrate an understanding of multiplication (2- or 3- digit by 1- digit) to solve problems by:

  • applying the distributive property.

N8

Demonstrate an understanding of fractions less than or equal to one by using concrete, pictorial and symbolic representations to:

  • compare and order fractions
  • model and explain that for different wholes, two identical fractions may not represent the same quantity

N10

Relate decimals to fractions and fractions to decimals (to hundredths).

 

Grade 5:

N4

Apply mental mathematics strategies for multiplication, such as:

  • using the distributive property.

N7

Demonstrate an understanding of fractions by using concrete, pictorial and symbolic representations to:

  • create sets of equivalent fractions
  • compare fractions with like and unlike denominators.

N9

Relate decimals to fractions and fractions to decimals (to thousandths).

N10

Compare and order decimals (to thousandths) by using:

  • benchmarks
  • equivalent decimals.

 

Grade 6:

N4

Relate improper fractions to mixed numbers and mixed numbers to improper fractions.

N5

Demonstrate an understanding of ratio, concretely, pictorially and symbolically.

P/R5

Demonstrate and explain the meaning of preservation of equality, concretely and pictorially.


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Evidence of Learner Understanding

What level of understanding do your students have regarding Equality?  Use the “Big Ideas of Equality”  to guide your teaching.  Sample evidence is provided for each Big Idea in order to guide you when assessing your students’  level of understanding.

Big Idea of  Equality What Might It Mean?      What might be evidence of understanding?
Student can match quantities without counting. In primary: Students are expected to subitize quantities of 2 , 3 and 5. In upper elementary we expect they see equivalence with fractions and decimals as automatic. Students match small sets. They may physically match one to one. They might pair objects. They might just look and know.

Matched the quantity accurately.

Student does not need to count or prove by counting. Simply agree that these are equal because they are the same amount .

Attributes of objects .
Student is able to explain why sets whose objects have been matched have same number: one to one correspondence, automatic recognition of parts or wholes or both (subitize, part whole relationships), conservation of quantity. Matching the pattern or arrangement

Students are expected to discuss and explain how they know these sets are equal.

I lined them up and they match, there is one for each.

I counted and they are the same amount.

I saw they fit the same space.

They are arranged the same.

They are the same length.

I moved them in my head, they are the same.

Demonstrate a strategy, describe a strategy, create an equation and diagram the strategy. (Number Lines, strings, folding, covering equal spaces.

These fit on each other.

These line up.

These fill the same container to the same point.

These hold the same position on the numberline “making ten” strategy with ten frames

Eg

  • 7 + 8 = 5 + 2 + 8 = 5 + 10
  • 15 is the same as 10 + 5; 8 + 7; 9 + 6; 2 + 13
  • 37 is the same as 20 + 17; 30 + 7; 35 + 4
  • 457 + 243 is the same as 460 + 240
Student recognizes rearranging does not change the equivalence

Equality is set up. Students agrees.

Items on either side are re-arranged, moved apart, traded across or substituted but if the quantity does not change the equality is not affected.

Are these equal?  Show picture, show equation, show expression

Does the equal sign fit here?

What do I have to do to “balance it”

Here is an equality, Build and Explain why it is true or “balanced”

Change it but keep it balanced.

7 = 5   no     add 2 or subtract 2 or move one over

3  + 4  = 7   yes   could change both sides and still be equal as in 4 + 3 = 2 + 5  7 = 3 + 4  etc.
Student recognizes that number can be an attribute used to sort sets.

These go together because they all have 5 in them. These go together because they are all ways to demonstrate 6.

These all equal 1/2.

These are all ways to show 1

Equivalent fractions, equivalent areas, equivalent volumes

Students identify equality based on same number of things.

Students replace the stuff with numbers and then confidently manipulate those numbers

These are acceptable equations about equal:

  •  5 = 5  ( there does not have to be an operation sign)
  • 2 + 3 = 5
  • 5 = 2 + 3
  • 2 + 3 = 3 + 2
  • 1 + 4 = 2 + 3

Student identifies sets that are equal and identifies why: contain same amount (strictly a count), or share a common measure as in equal weight, equal length, equal capacity or volume.

Student creates an equality to match a given equation.

Student identifies sets that are not equal with explanation for why.
Student holds attribute(s) firm. Objects used do not sway thinking. These are both sets of 8 even though one set is a different colour or a different object.

I used bears and houses but 5 is five.

One half of a chocolate bar and one half of an elephant. Both represents halves.... do not let the object interfere.

75 % is three quarters of something, does not matter what the something is...

The difficulty here is that when we are just talking number, objects can interfere. When we quantify with weight or length as in 3 cm or 5 gm... now it is a clearer image for many.

Thirds can be cut into sixths so two thirds is now four sixths.

Students identify equality based on same number of things.

Students replace the stuff with numbers and then confidently manipulate those numbers

Recognize that when two sets (collections) contain the same amount or quantity they are equal or balanced regardless of the arrangement or composition, colour, shape of the collection or set.

Student uses appropriate language correctly:  equal, not equal, more, less, equivalent, commutative, equality, inequality, equal sign, greater than, less than, same amount

Student records the appropriate statement of equality or inequality

Do not introduce symbols without purpose.. The greater than , less than symbols cause great confusion when students do not experience enough with contexts and materials.

The equal sign always means equal....

The convention we follow: one equal sign per equation, but we make exceptions when we are stringing equivalencies.

3+7=8+2=10 is true across both equal signs

3+7 = 10+4 = 14 is not.

6 = 2 + 4 os just as valid as 2 + 4 = 6 (You do not have to turn it around)

( Teachers worry that this is confusing and leads students to confusions around subtraction equations. Subtraction is a problem when we only focus on writing equations. When we build subtraction, explain and link the writing of equations to the explanations it is not a problem.

This same problem holds true for division when all we focus on is writing equations.

Expression  3 + 4 (student might see this in dot collection)

Expression 30 + 5 (student might see this in hundred grid)

Expression 5 + ? (students might set this up as they interpret a problem)

Expression 6b (Grade 5 we start to consider coefficients)

Equation:

6 = 4 + 2    or 4 + 2 = 6

7 = 9 – 2 or  9 – 2 = 7

3+5=5+3

8+2 = 7+3

6x4=4x6

6x4=24  or 24=6x4

24 ÷ 4 = 6   or 6 = 24÷ 4

4 + ? = 10  10 – 4 = ?

y = 2x + 1   y+5 = x + 2  

 

These are acceptable equations about equal:

 5 = 5  ( there does not have to be an operation sign)

2 + 3 = 5 5 = 2 + 3   2 + 3 = 3 + 2   1 + 4 = 2 + 3

Student identifies sets that are equal and identifies why: contain same amount or share a common measure as in equal weight, equal length, equal capacity or volume.

Student creates an equality to match a given equation.

Student identifies sets that are not equal with explanation for why.
Student accepts and applies the term equal to quantifiable comparisons as in equal weights, distances, time, areas, volumes, etc.

As we move to Grade 3 we begin to introduce units of measure. Units of measure are related. 10 mm = 1 cm  100cm= 1m

Place Value units are equivalent: 10 tens = 100

10 hundreds = 1000

Fractional Units and Decimal Fraction Units are embedded

Thirds are in sixths hundredths are in tenths
Students recognize that changing units does not change the value of the measurement.

Ex. Changing 1 km to 1000 m does not make it longer or shorter.
Student is able to preserve equality between 2 sets. (Students recognize that equality is a relationship that can be conserved.) Students recognize that you can maintain equality by adding, subtracting, multiplying and dividing both sides by the same amount.

Students identify equality based on same number of things.

Students replace the stuff with numbers and then confidently manipulate those numbers

Student is able to transform inequality to create equality multiple ways and can explain the difference between those ways and record equations to describe

Addition and Multiplication

Student is able to use multiple ways to transform an equality into an equivalent equality for both addition and multiplication.


Liping Ma "Changing one or both sides of an equal sign for certain purposes while preserving the "equals" relationship is the secret of mathematical operations".

Students demonstrate flexibility in number sense.

Students can apply the big ideas of equality to facilitate the learning of basic facts

Liping Ma “Changing one or both sides of an equal sign for certain purposes while preserving the “equals” relationship is the secret of mathematical operations”.

Array for multiplication:

  • 8 x 3 = 3 x 8(commutative)
  • 3 x 8 = (1 x 8)+(2 x 8)(decomposing)(distributive)
  • 3 x 8 = 6 x 4(double and half)

  • If  5 + 5 = 10 then 6 + 4 = 10(rearranging does not change equivalence)
  • If  5 + 5 = 10 then 5 + 6 = 11(preservation of equality)

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Research Links

ResearchWhat researchers say about equality:

  1. Students in elementary school often misinterpret the equal sign (=) as an operational (i.e., do something or write an answer) symbol even though the equal sign should be viewed as a relational symbol (Sherman & Bisanz, 2009). Students should understand the equal sign as relational, indicating that a relationship exists between the numbers or expressions on each side of the equal sign (Jacobs, Franke, Carpenter, Levi, & Battey, 2007). The number or expression on one side of the equal sign should have the same value as the number or expression on the other side of the equal sign. If the equal sign is interpreted in an operational manner, this typically leads to mistakes in solving equations with missing numbers (e.g., 5 − ___ = 1) and difficulties with algebraic thinking (e.g., x − 2 = 2y + 4; Lindvall & Ibarra, 1980; McNeil & Alibali, 2005b). Research has shown, however, that ongoing classroom dialogue (e.g., Blanton & Kaput, 2005; Saenz-Ludlow & Walgamuth, 1998) or explicit instruction (McNeil & Alibali, 2005b; Powell & Fuchs, 2010; Rittle-Johnson & Alibali, 1999) can change students’ incorrect interpretations of the equal sign.

    This misconception, according to researchers at Texas A&M University, can inhibit a student’s mathematical achievement. According to their research, students who exhibit the correct understanding of the equal sign show the greatest achievement in mathematics and persist in fields that require mathematic proficiency like engineering (Capraro & Capraro 2010).

 

  1. TEACHING COMPUTATION WITHOUT ATTENTION TO RELATIONAL AND ALGEBRAIC THINKING ERECTS A ROADBLOCK TO STUDENTS’ LATER PROGRESS. Students must see all math as a search for patterns, structure, relationships, as a process of making and testing ideas and in general making sense of quantitative and spatial situations (Schoenfild, 2008)

 

  1. Liping Ma “Changing one or both sides of an equal sign for certain purposes while preserving the “equals” relationship is the secret of mathematical operations”.

 

  1. Equivalence obeys three properties:

    1. reflexivity, the relationship relates a thing to itself,  A is related to A;

    2. symmetry, if A is related to B, then the opposite direction is also true, i.e; B is related to A; and

    3. transitivity, if A is related to Band B is related to e, then A is related to C.

Lecture

  • The most used equivalence relation in mathematics is 'equals'.

  • For the equivalence properties, transitivity appeared to be well understood, symmetry reasonably understood and reflexivity not recognized.

  • Equality relationships were commonly seen in terms of balance and students were skilful in returning .relationships to balance when they were unbalanced;

  • Equality as transformation was less familiar and older students appeared unable to interpret examples such as x+3=5 in terms of change and reversing change, even' when an example was worked through with them..

  • Researchers found NO EVIDENCE of recognition of reflexivity. Cooper, Rixon, Burnett

 

More Readings:

Powell, S. R. (2012). EQUATIONS AND THE EQUAL SIGN IN ELEMENTARY MATHEMATICS TEXTBOOKS. The Elementary School Journal, 112(4), 627–648.

Eric J. Knuth, Martha W. Alibali, Shanta Hattikudur, Nicole M. McNeil, and Ana C. Stephens (2008).The Importance of Equal Sign Understanding in the Middle Grades. (PDF)

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Try This

Quick Assessment Rubric

Equality Quick Assessment

Original version - PDF

Original version - Word

These activities were provided in the Equality webinar.  Choose at least one activity to give to your students.  Have them record their responses.  Pay special attention to how students get to their answers.  Do they have an operational or relational understanding of Equality?

Equality - Addition

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Equality - Multiplication

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Equality - Research Question

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Resources

Programming decisions and the selection of learning resources are made by school authorities, schools, teachers and students.

The use of authorized resources is not mandatory.  A broad range of learning resources may be used to meet the needs of all students.

The links below will provide you with a variety of resources that may come in handy when teaching the concept of equality.  


These resources are suggestions and are meant to complement what you are already using.  They are not necessarily aligned to Alberta Curriculum. These resources come from a variety of sources and are not affiliated with Alberta Education.

Click here to submit a resource

Kindergarten: NUMBER

2. Subitize (recognize at a glance) and name familiar arrangements of 1 to 5 objects or dots.

4. Represent and describe numbers 2 to 10, concretely and pictorially.

5. Compare quantities 1 to 10, using one-to-one correspondence.

Kindergarten

Grade 1: NUMBER

2. Subitize (recognize at a glance) and name familiar arrangements of 1 to 10 objects or dots.  [C, CN, ME, V]

4. Represent and describe numbers to 20, concretely, pictorially and symbolically.

5. Compare sets containing up to 20 elements, using:

      • referents
      • one-to-one correspondence to solve problems.

7. Demonstrate an understanding of conservation of number.

Resources - Grade 1: NUMBER

Grade 1:  PATTERNS AND RELATIONS (Variables and Equations)

4. Describe equality as a balance and inequality as an imbalance, concretely and pictorially (0 to 20).

5. Record equalities, using the equal symbol.

Resources - Grade 1: PATTERNS AND RELATIONS (Variables and Equations)

Grade 2: NUMBER

5. Compare and order numbers up to 100.

Resources - Grade 2: NUMBER

Grade 2: PATTERNS AND RELATIONS (Variables and Equations)

4. Demonstrate and explain the meaning of equality and inequality, concretely and pictorially.


5. Record equalities and inequalities symbolically, using the equal symbol or the not equal symbol.

Resources - Grade 2 : PATTERNS AND RELATIONS (Variables and Equations)

Grade 3: NUMBER

3. Compare and order numbers to 1000.

13. Demonstrate an understanding of fractions by:

      • explaining that a fraction represents a part of a whole
      • describing situations in which fractions are used
      • comparing fractions of the same whole that have like denominators.

Grade 3 : NUMBER

Grade 4: NUMBER

2. Compare and order numbers to 10 000.

6.  Demonstrate an understanding of multiplication (2- or 3- digit by 1- digit) to solve problems by:

      • using personal strategies for multiplication with and without concrete materials
      • using arrays to represent multiplication
      • connecting concrete representations to symbolic representations
      • estimating products
      • applying the distributive property.

8. Demonstrate an understanding of fractions less than or equal to one by using concrete, pictorial and symbolic representations to:

      • name and record fractions for the parts of a whole or a set
      • compare and order fractions
      • model and explain that for different wholes, two identical fractions may not represent the same quantity
      • provide examples of where fractions are used.

10. Relate decimals to fractions and fractions to decimals (to hundredths).

Grade 4: NUMBER

Grade 5: NUMBER

4. Apply mental mathematics strategies for multiplication, such as:

      • annexing then adding zero
      • halving and doubling
      • using the distributive property.

7. Demonstrate an understanding of fractions by using concrete, pictorial and symbolic representations to:

      • create sets of equivalent fractions
      • compare fractions with like and unlike denominators.

9. Relate decimals to fractions and fractions to decimals (to thousandths).

10. Compare and order decimals (to thousandths) by using:

      • benchmarks
      • place value
      • equivalent decimals.

Grade 5: NUMBER

Grade 5 : PATTERNS & RELATIONS

(End of page 2)

Grade 5 : PATTERNS & RELATIONS

Grade 6: NUMBER

4. Relate improper fractions to mixed numbers and mixed numbers to improper fractions.

5. Demonstrate an understanding of ratio, concretely, pictorially and symbolically.

Grade 6: NUMBER

Grade 6 : PATTERNS AND RELATIONS

3. Represent generalizations arising from number relationships, using equations with letter variables.

      • Develop and justify equations using letter variables that illustrate the commutative property of addition and multiplication; e.g., a + b = b + a or

          a × b = b × a.

5. Demonstrate and explain the meaning of preservation of equality, concretely and pictorially.

Grade 6 : PATTERNS AND RELATIONS

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Parent Communication

Ideas for Using the Following Information:

The information included below can be used in multiple ways.

  • Portions can be included monthly parent bulletins/newsletters

  • Items can be discussed at Parent-Teacher Interviews

  • Ideas can be used in a Parent Information Night



Message 27 - A Math Message to Families (Seeley, 2009)

http://mathsolutions.com/documents/9781935099031_message27.pdf

NCTM - Support for Families - Helping Your Math Students

http://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/J_-Michael-Shaughnessy/Support-for-Parents-and-Families_-Helping-your-Math-Students/

NCTM - Algebra - Connect it to Student Priorities!

http://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Henry-(Hank)-Kepner,-Jr/Algebra%E2%80%94Connect-it-to-Students_-Priorities!/


Newsletter

Alberta Education’s Parent resource area:

http://education.alberta.ca/teachers/program/math/parents/links.aspx

Big Ideas

What is equality?  Learning

  • Equality is when you have an equal number of items in two groups



Why is it important?  Children need a strong understanding of equality.  It allows them to

  • compare quantities and decide if one group has more, less or is equal.  

  • move from arithmetic to algebra

 

Simple Activities you can do with your child

Download in PDF | Download in .DOC

Activity 1
    • Place some toys in front of your child.  Ask them how many there are.  Rearrange the items without adding or removing any.  Ask them how many there are.

    • Why?  This helps them see that just because you’ve moved the toys around, the quantity doesn’t change.

    • Split the toys up into two piles that are close together.  Did the quantity change if I’m looking at all of the toys in front of me?  No.  7 toys broken up into a pile of 2 and a pile of 5 is still 7 toys.

    • Trade out one of your toys for different toy.  Did the quantity change?  No.  If I started with 7 and trade one of those toys for a different toy, I still have 7 toys.

    • Repeat with other objects.

    • NOTE:  Students need to see that moving objects around (without adding or removing anything) doesn’t change the quantity of the items.  4 items shuffled around will still be 4 items.

Activity 2
    • Give your child some toys.  Give yourself the same number of toys.  Ask if the groups are equal.  

    • Rearrange the toys on your side.  Ask if the groups are equal.

    • Repeat for their toys.

    • Split your pile up into two small piles.  (Ex. you both have 7 toys.  You split your toys up into a pile of 3 and a pile of 4).

    • Indicate both your piles.  Tell them that you have a pile of 3 and a pile of 4.  Do you have the same number of toys as they do?

    • Ask them “Is a pile of 3 and a pile of 4 the same as a pile of 7?”

    • Combine your piles back into one big pile.  Have them split their pile into 2 piles and ask the question again.

    • Both of you break your big pile into two piles and compare.

    • Repeat with other objects.

    • Repeat with other objects but make your pile slightly different than your child’s.  Examples:  

      • You have cheerios.  Your child has fruit loops.

      • You have apples.  Your child has oranges.

      • You have chocolate bars.  Your child has grapes.

    • NOTE:  A major misconception that children have is that the answer comes after the equals sign.  (Example:  4 + 3 = 7)  Develop their understanding of equality by switching it so that the “answer” is before the equals sign (Example:  7 = 4 + 3),

Activity 3:  Moving to Abstract
  • Write out one of the problems you just did.  For example, a pile of 3 toys and a pile of 4 toys is the same as a pile of 7 toys.

  • Complain that this is a lot of writing.

  • Write down the number 3 below “a pile of 3 toys” and ask if this makes sense.

  • Ask them if they know a mathematical way of writing “and”.  (Note:  Depending on the age/experience level of your child, you may have to just show them that “and” can be written as “+” in this case).  

  • Write the “+” sign below “and”.

  • Ask what you could write instead of “a pile of 4 toys”.  Write 4 below it.

  • Explain that we use “=” to show “is the same as” and write this below “is the same as”

  • Ask what you could write instead of “a pile of 7 toys”.  Write the 7.

  • Reread the equation you should wrote.  “3 + 4 = 7” as “a pile of 3 toys and a pile of 4 toys is the same as a pile of 7 toys”.

  • Recombine your two piles into one big pile.

  • Have your child split their pile into two piles.  (ex. 2 + 5)

  • Say the long way “one pile of seven toys is the same as one pile of two toys and one pile of five toys.”

  • Write out the equation.  7 = 2 + 5

    • Note:  make sure to put it in this format.  Children who see the equation written both of these ways will have a deeper understanding of equality.

  • Repeat this activity but this time both of you split your piles.  For example 1 + 6 = 4 + 3

  • NOTE:  A major misconception that children have is that the answer comes after the equals sign.  This activity will help students understand that the equals sign means that whatever is on the left side must be the same as whatever is on the right side.

Activity 4

    • Start with two equal piles again.

    • Explain that you are going to split yours up into two piles.  You’re not allowed to add any or take any away.

    • Ask your child to close their eyes.

    • While their eyes are closed, split your pile into two and cover one pile.

    • When they open their eyes, ask them to tell you how many you covered.  

      • Example:  3 + ??? = 7

    • Have them do the same for you.

      • Example:  7 = ??? + 1

    • Repeat but this time split both sets of piles but cover only one pile.

      • Example:  1 + 6 = ??? + 2

    • NOTE:  A major misconception that children have is that the answer comes after the equals sign.  So when they see 1 + 6 = ___ + 2, they will usually say that ___ is 7.  Then + 2.  Therefore the answer is 9.  This activity will help students understand that the equals sign means that whatever is on the left side must be the same as whatever is on the right side.

Activity 5
    • Create two equal pile of toys.  

    • Add a few more toys to your pile.  (Example:  2 toys)

    • Ask your child “What would you have to do to make your pile the same as my pile?”  They should say “Add 2 toys).  Have them do this so your piles are even again.

    • Your child now adds some toys to their pile.  Example:  3  You say “Now I have to add 3 toys to my pile so that my pile is the same as your pile.  Do it.

    • This time, remove some toys from your pile.  Have them make the piles even.

    • Show what this looks like mathematically.

      • Example:  You both started with seven toys so write 7 = 7

      • You add 2 toys and they add 2 toys:  7 + 2 = 7 + 2

      • You are now starting with 9 = 9

      • They add 3 toys to both sides so 9 + 3 = 9 + 3

      • You are now starting with 12 = 12

      • You remove 4 toys.  12 - 4 = 12 - 4

    • NOTE:  This will help students develop a sense of “Preservation of equality”.  Whatever you do to one side must be done to the other side in order to keep both sides equal.


How equality changes from year to year?

  • In Kindergarten

    • an understanding of equality starts with students recognizing and naming familiar arrangements of 1 - 5 objects or dots (this is called subitizing)Classroom

    • students compare two groups of items (groups from 1 - 10) by matching up the items in each group.  This allows students to talk about which group has more items or less items or do the two groups the same number of items.

 

  • In Grade 1

    • Students will recognize, name and write numbers 1 - 20, as well as being able to draw objects showing the number (for example the number 7 has 7 objects to draw out), and if they have items (like beads) then students will show that the number 16 means there are 16 beads.

    • Students continue to compare two groups of items, and now the groups may have 1 - 20 items in each group.   

    • Students will talk about the groups, if each group has the same number of items, the groups are “balanced”, if one group has more or less, then the groups are “imbalanced”

    • Students will be able to look at groups of items and write the math equation that shows what is happening with the groups.  For example, if you have one group with 7 items, and then another group with a pile of 3 and a pile of 4 items, the student will be able to write:    7 = 3 + 4

    • If you were to switch the piles around in a different order, for example if you had the first group with 7 items, and then in the second group you changed the piles into a pile of 2 and a pile of 5, the student would know that the two groups are still equal (still have the same number of items)

Misconceptions

As a reminder, the equal sign shows that two groups have the same number of items.

The Question:  

(Focusing on the Misconception)
The Follow Up Question The Background Info to be Provided

The answer is always on the right of the equals sign.  

Ex. 3+5=?
?=3+5

Misunderstanding:  Students see this as wrong.  They feel the answer must come after the equals sign.


Since the equal sign is showing that two groups have the same number of items, it does not matter which group you look at first.


3+ 5 = 8     is showing the same relationship as


8 = 3 + 5

An equal sign means the answer is coming after.  

Ex. 3+5 = ?
3+5=?+2

Misunderstanding:  Students often answer this as 3+5=8+2  


Children see the equals sign as an operation rather than as a relationship.


Suggestion:  Use the phrase “is the same as” and other phrases rather than just “is equal to” when referencing the equals sign

Vocabulary

Parent Friendly

Equality

  • Equality is when you have an equal number of items in two groups

  • Students are comparing the number of items in two groups.  When each group has the same number of items, you have equality

Inequality

  • Inequality is when you have different numbers of items in two groups

  • One group has more or less than the other group

Attribute

  • Attributes are ways of describing items, such as:

    • Quantity (how many)

    • Colour

    • Length

    • Weight

    • Shape

Conservation of number

  • Conservation of number is when a student knows that the number stays the same even when it looks different.

  • For example,


Addition   is the same amount as


Addition




or  3 + 2  is the same as 2 + 3

Symbolically

  • Symbolically writing math is when students use written numbers and math symbols (not pictures, and not real objects)

  • For example:   13 + 8 = 21

Concretely

  • Representing a situation or solving a problem concretely means using actual objects (for example, bingo chips, beads, or blocks, etc.)

Pictorially

  • Working on math pictorially means using drawings or pictures

Equation

  • An equation shows two groups that are equal in number (even if they are written in a way that makes them look different).  Note: One group is shown before the equal sign and one group is shown after the equal sign.

  • Examples of equations include:

    • 4 + 7 = 11

    • 9 = 7 + 2

    • 2x + 3 = 13

    • 5 + 3 = ____ + 6

Variable

  • A variable is something that is unknown, or something that can change its value.  Variables are usually shown as blank lines, empty boxes or letters.

  • Examples of variables are shown in pink below:

    • 2 + = 8

    • 5 + 3 = ____ + 6

    • 2x + 3 = 13

    • y = 3x - 5

Preservation of equality

  • If you start with two equal groups, these groups will always remain equal to each other if you do any of the following:

    • Add the same amount to both groups

    • Remove the same amount from each group

    • multiply each group by the same number

    • divide each group by the same number



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Last modified: Thursday, 15 September 2016, 2:06 PM