Hunting for Patterns and Rules

# 3. Equivalence

Suggested Learning Intentions

• To understand that expressions on either side of an equal sign must be balanced
• To explain that when a function on one side of an equation is performed an equal function must be performed on the other side to maintain balance

Sample Success Criteria

• I can articulate that an equals sign represents a balance of the expressions on either side of it
• I can create equivalent number sentences/equations
• I can manipulate equations or number sentences to maintain the balance on both sides
• I can model my problem solutions using a variety of manipulatives
• Algebra: Equivalence handout: PDF
• Thinkboard: docx PDF

Ask students, ’What does equivalence mean?’

Tell them that we use an ‘equals’ sign to show equivalence, which means that whatever is on either side of an equal sign must be balanced.

1. Five Blocks

Provide students the activity sheet from Lesson Two of reSolve’s Algebra: Equivalence sequence – Five Blocks. (The full sequence is available here: Algebra - Equivalence.)

Give them five minutes to look at the activity on their own. Then ask them to work with a partner to work out the balance problems.

When completed, call on different students to explain how they worked it out, and how they knew that both sides were or were not equivalent.

2. Balancing Numbers Card Game

In the same pairs, have students play a ‘Balancing Numbers’ card game (see image below). Each pair will need a pack of playing or number cards and a piece of paper.

Steps:

1) Draw four places for cards and add in operations as shown below.

2) Place a stack of five cards face down on each place.

3) Students take turns to remove one card and then flip over the next card in the pile.

4) The aim is to try to make the expressions equivalent.

Equivalent Number Sentences

Tell students: ‘Now that you have had some time to practice balancing two sides of a balance scale and of an equals sign, we will now explore working with more equivalences’.

Discuss with students that the parts on either side of the equal sign are called expressions. Any expressions which also include an ‘equal’ sign is called an equation.

1. Equivalence with Addition & Subtraction

Working in pairs, present the students with the following task (adapted from Making Both Sides Equal, Sullivan, 2018, p 24).

898 + ___ = 900 + ___

224 - ___ = ___ - 10

Say to students, ‘Please work out some numbers that make these equations true and give a range of possibilities for each equation. Please also show how you worked out each of the possibilities.’

Enable students by using more accessible numbers, based on their learning needs. For example:

9 + __ = 10 + ___

53 - __ = 74 - ___

Extend students by asking them to describe the patterns that explain their answers.

• How did you make the equations equivalent?
• What functions/operations did you use?
• Could you have worked it out another way?

2. Equivalence with Multiplication & Division

Present the students (in pairs) the following task – Missing Number Multiplication & Division (P Sullivan, 2018, p 44):

I completed a multiplication question correctly on the computer, but my printer ran out of ink. Now the question looks like this:

2_ × 3_ = _ _0

What might the digits that did not print be? Give as many possibilities as you can.

Enable students by asking, ‘What might be the missing numbers?’

_ _ × _ =_0

1_ × _ = _0

Extend students by asking them to convince you that they have all the possible solutions.

• How did you make the equations equivalent?
• What functions/operations did you use?
• Could you have worked it out another way?

### Areas for further exploration

1. True or false

This activity is based on True or False, adapted from Van De Walle et al, (2019, p 328).

Write these equations (or similar ones) on cards:

In pairs or small groups ask the students to sort these cards into two piles – one for True and one for False. Once completed, ask the students the following questions:

1. How do you know that the statements are true or false?

2. What would you need to do to make the false statements true?

The aim of this stage is that students will understand that the ‘equal’ sign means ‘balance’. Both sides of an ‘equal’ sign must be equivalent. Understanding and seeing the patterns, along with using operations/functions will assist students in balancing equations.

Tell the students to look at the following equation:

68 + 34 = 52 + __

• What do you need to do to balance this equation?
• How would you go about working out what the missing number is?
• What functions/operations did you use?
• If you added 2 to the 68 to make this expression easier to work with, what do you need to do to the expression on the other side for the equation to continue to be balanced?
• Did you notice any patterns or rules?

To assess students’ understanding of Equivalence, provide students with a Thinkboard (available as a handout from Materials and texts).

Australian Academy of Science, n.d. Algebra: Equivalence. [Online]
Available at: https://www.resolve.edu.au/algebra-equivalence
[Accessed 15 March 2022].

Australian Association of Mathematics Teachers, n.d. 4 Arm Shapes. [Online]
Available at: maths300.com/members/m300full/040l4arm.htm
[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

Melbourne Archdiocese Catholic Schools, 2013. Key Ideas for Concept Development in Mathematics. East Melbourne: Melbourne Archdiocese Catholic Schools.

Nathan, M. & Knuth, E., 2003. A study of whole classroom mathematical discourse and teacher change. Cognition and instruction, 21(2), pp. 175-207.

Parsons, J. & Reilly, Y., 2012. Meet the Johnsons. Maths in the Inclusive Classroom Book 1. s.l.:Essential Resources.

Philips, E., Gardella, T., Kelly, C. & Stewart, J., 1991. Patterns and Functions, Addenda Series, Grades 5-6. Reston, VA: NCTM.

Siemon, D. et al., 2015. Teaching Mathematics: Foundations to Middle Years. Melbourne: Oxford University Press.

State Government of Victoria, (Department of Education and Training), 2017. Monster choir: making patterns. [Online]
Available at: https://fuse.education.vic.gov.au/Resource/LandingPage?ObjectId=e066b087-7026-479e-9f9e-7b1f4b0422c1&SearchScope=Primary
[Accessed 15 March 2022].

State Government of Victoria, (Department of Education and Training), n.d. Introduction of variables using letters to represent numbers. [Online]
Available at: https://fuse.education.vic.gov.au/mcc/CurriculumItem?code=VCMNA251
[Accessed 15 March 2022].

Stein, C., 2007. Promoting mathematical discourse in the classroom. NCTM, vol 101, no.4. [Online]
Available at: http://hybridalgebra.pbworks.com/f/lets+talk+promoting+mathematical+discourse+in+the+classroom.pdf
[Accessed 15 March 2022].

Sullivan, P., 2017. Challenging Mathematical Tasks. South Melbourne: Oxford University Press.

TopMarks, n.d. Function Machines. [Online]
Available at: https://www.topmarks.co.uk/Flash.aspx?f=FunctionMachinev3
[Accessed 15 March 2022].

Toy Theatre, 2001-2022. Number Pattern. [Online]
Available at: http://toytheater.com/number-pattern/
[Accessed 15 March 2022].

University of Cambridge, n.d. NRICH: Got It. [Online]
Available at: https://nrich.maths.org/1272/index
[Accessed 15 March 2022].

University of Cambridge, n.d. NRICH: Mathematics Resources for Teachers, Parents and Children. [Online]
Available at: https://nrich.maths.org/
[Accessed 15 March 2022].

University of Cambridge, n.d. NRICH: Sticky Triangles. [Online]
Available at: https://nrich.maths.org/88/index
[Accessed 15 March 2022].

Uttal, D., Scudder, K. & DeLoache, J., 1997. Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of applied developmental psychology, 18(1), pp. 37-54.

Van de Walle, J., Karp, K., M, B.-W. J. & Brass, A., 2019. Primary and Middle Years Mathematics: Teaching Developmentally. Australia: Pearson.