Unknowns and variables: Solving one-step equations

Purpose

The purpose of this unit is to support students as they read and record equations in which letters (or variables) represent an ‘as yet unknown’ amount.

Specific Learning Outcomes
• Understand that an unknown amount or number can be represented with a symbol: a question mark, a shape or a letter.
• Recognise that to find the value of the variable, you have to ‘undo’ what has been done to it.
• Write word problems of real life situations and express these with equations that include a variable.
• Recognise that an equation is balanced around the equals symbol.
• Formally solve equations, which include variables and which represent problem situations involving fractional and decimal amounts.
• Estimate values for unknown amounts and explain reasoning.
• Recognise the calculator is a useful but ‘fallible’ tool and that errors are possible and give explanations for these.
Description of Mathematics

The focus of learning continues to be on developing understanding of the symbols that we use to express our mathematical ideas and to communicate these ideas to others. In earlier units, students have been learning the meaning of symbols, have come to understand the relationships between operations, and have come to recognise the role of number properties in solving problems. Students have been helped to see that symbols, expressions and equations are thinking, and writing tools that we use to express and help solve real life problems.

Students have been solving problems involving unknowns for as long as they have been ‘operating on numbers’. 2 + 3 = ☐, 2 + ☐ = 5 and ☐ + 3 = 5, are problems students encountered when they solved problems by counting from one on materials. In this unit the transition between representations of an unknown amount is made from the ‘box’ ☐, to the pro-numeral or variable. In introducing and exploring the variable, the most basic tool kit of symbols that we use to express a problem, or to describe a (linear) relationship, is now assembled.

Students need to have developed important understandings and ways of thinking, in order to understand and use variables. In developing algebraic thinking, students are more focused on relationships; in particular, they have come to understand that in an equation, the relationship is one of equivalence. Strong relational thinking is demanded of the student as they develop the important conceptual view of an equation as ‘an object that balances’.

As students work to find the value of the unknown, they come to understand that they are in fact, ‘getting the variable by itself’, or ‘isolating the variable’. In doing so they apply inverse operations and properties such as the additive inverse. Understanding these processes means that, in solving problems, they can do so formally and with understanding, rather than simply following a taught procedure. It is an important culmination of algebra skills and understandings when students can, with understanding and confidence, read, write and solve one step equations involving variables, which represent real life problems.

Links to the Number Framework

Required Resource Materials
balance scales

plastic cubes

Activity

Ordering items from a school canteen menu is the context for this unit.

There are four sessions only in this unit. It is important, however, that the number of sessions is increased, if appropriate, to ensure the concepts are fully understood. It is also important to ensure that quality time is given to discussions, with student explanations being encouraged and valued.

Session 1

SLOs:

• Understand that an unknown amount or number can be represented with a symbol: a question mark, a shape or a letter.
• Write expressions using variables.

Activity 1

1. Begin the session by discussing lunch options in your school.
Display Attachment 1, the canteen menu for Kiwi School. Read it together checking prices of random specific items using the price list in the box at the bottom of the menu.
Note that some differences in daily special prices may reflect the size of portions and that only some students buy their food from the canteen.
Write on the class modelling book:
\$3.00 + \$3.00 + \$3.00 + \$3.00 + \$3.00 + \$3.00 + \$3.00 + \$3.00 = \$24.00.
Ask what is a quicker way to write this?
Agree on \$3.00 x 8, or 3 x 8 = 24 (dollars)
\$24 is spent on \$3.00 orders. How many orders were placed?
Agree 24 ÷ 3 = 8 finds the number of \$3.00 orders placed.
Ask, “What is the relationship between multiplication and division?”
Agree that it is an inverse relationship.
2. Make Attachment 2a and pencils available to each student.
Explain that the students in Room 7 at Kiwi School have the responsibility for lunch orders and that there are details missing from Monday’s order form.
Have the students complete the form either individually or in pairs, then share and discuss their results.

Activity 2

1. Discuss the strategies student used to complete the form. Write examples of equations, highlighting that when they were multiplying they were finding the value of unknown amounts.
Write,
Vege quiche: 4 x = 28
Tuna wrap: x 6 = 30
Cheese rolls: 2 x 15 =
Discuss the symbol ☐, and how they have seen and used this symbol often.
Ask, “What else could we write to show an unknown (amount)?
Record suggestions and rewrite the equations using these. For example:
4 x ? = 28         x 6 = 30          2 x 15 = n
2. Highlight that all of these can be used to ‘stand for’ or ‘represent’ a number.
When we solve equations with symbols for unknown amounts we have to find out what number the symbol stands for.
Make pencils and paper available.
Pose this problem for student pairs to discuss and answer:
What numbers could the symbols for the unknowns in these equations stand for? 3. Have students pair-share their solutions. Encourage the students to use the language: “the possible values for the unknowns are…”
Write any new vocabulary in the modelling book or on a poster to refer back to.
Discuss ideas as a class, highlighting these key points:
• The different shapes represent different values, therefore the addends cannot be 7 + 7 + 7
• The triangle and square represent factors of 24.
• The letters ‘a’ and ‘b’ represent different values, therefore the value of the addends cannot be 7 + 7 + 4
• The only possible value for ‘y’ is 5 because the same symbol shows that the factors have the same value.
4. Write 3n on the class modelling book. Have students discuss and explain its meaning. Agree it means ‘3 times an unknown number’. Highlight the fact that the multiplication symbol does not need to be shown.

Activity 3

1. Make Canteen menus (Attachment 1), Attachment 2b and pencils available to each student.
Explain that the prices are on the canteen menu, but that the lunch orders for Tuesday are shown differently.
Ask what ‘n’ represents. (The number of items ordered). Encourage students to solve problems by posing multiplication questions like “4 times what is equal to 20?” ( 4 x ☐ = 20) and “3 times 5 is equal to what?” (3 x 5 = ☐)
Have the students complete the form and then pair share and discuss their results.
2. Discuss the strategies student used to complete the form. Write examples of equations for selected problems, using the order form context. For example:
“Fifteen dollars were paid for Chicken and Vege kebabs. How many were ordered? How do we know?”
Highlight the fact that when they were multiplying, they were finding the value of an unknown amount.
Write variable on the class chart, explaining that this is a term used for a number we don’t yet know.
Conclude the lesson by recording on the class modelling book and discussing the kinds of symbols we have been using: numerals, operations symbols (+ - x ÷), a relationship symbol (=) and variables (letters).
Agree that these symbols together make it possible to express our mathematical thinking and to solve problems. Recognise that sometimes algebra enables us to solve problems that can’t be solved in any other way.

Session 2

SLOs:

• Recognise that to find the value of the variable, you have to ‘undo’ what has been done to it.
• Write word problems of real life situations and express these with equations that include a variable.
• Solve single step whole number equations that include a variable.

Activity 1

1. Refer to the order forms for Monday and Tuesday, from Session 1. Review key learning from this session.
Write on the class modelling book n + 3. Ask students to discuss in pairs what this expression means (3 more than an unknown amount). Ask students to offer scenarios which the expression n + 3 could represent.
Accept and discuss a range of possible scenarios. For example:
On Wednesday a number of orders had been received for sushi. Then 3 more orders came in: n + 3 was the total number of orders.
2. Write n + 3 = 21. Ask: What was the original number of orders received?
Ask students to discuss in pairs what they need to do to find the value of ‘n.’
Agree that an addition or a subtraction question could be asked.
Record these: “What plus 3 is equal to 21?” ( ☐ + 3 = 21) or “21 minus 3 is equal to what?” (21 – 3 = ☐)
Ask, “What do you notice about these number operations?”
(They are inverse operations. Subtraction undoes addition.) We can solve this addition problem, finding the value of the variable, by subtracting.
3. Write 3n on the class modelling book and review this means 3 times an unknown amount. Discuss how the value of n can be found. Agree that the opposite operation of division (divide by 3) will ‘undo’ the multiplication and will give 1n or n.
Ask students to demonstrate two ways they can show this division and discuss these:
 3n ÷ 3 = 1n or n, 3n = 1n or n 3

Activity 2

1. Make Attachment 1 (Canteen Menu) and 2c, and pencils available to each student.
Point out that the order of the columns is different from Monday and Tuesday’s forms and that in each line they are finding the value for the variable, n.
Have students work in pairs to complete this order form, each recording their own working. Remind students that unit prices are on the menu and have them notice which operations they are using, recording these in the calculation column in two ways.
2. Have students pair-share their work and discuss their results.

Activity 3

1. As a class group discuss and record the total amount spent on canteen lunches at Kiwi School on Monday (\$135), Tuesday (\$94) and Wednesday (\$119).
Explain that Thursday is usually the day on which orders are the highest, so the numbers will be (a little) larger.
Have students suggest order scenarios and beside these record equations that express these.
For example:
Altogether forty-two dollars were spent on toasted chicken sandwiches, for which the price is three dollars each. How many toasted chicken sandwiches were ordered.
42/ 3 = n
How many vege nachos were ordered, at four dollars each, if seventy-two dollars were spent? 4n = 72
On Thursday there were seventeen orders for fresh fruit. This was five more than on Wednesday. How many orders were placed on Wednesday? n + 5 = 17
2. Make pencils and paper available.
Have each student write at least five (word) scenarios of their own, also expressing these with equations in which n represents the number of items.
They should show their solutions to their own equations.
3. Have students read their scenarios to a partner, and have this person write the equation that would express the word scenario. The student pair should then compare their equations.
4. Have some students share one of their scenarios with the class.
Reflect as a class on the importance of the inverse operation in solving the problems they have written using a letter to represent an unknown amount.

Session 3

SLOs:

• Recognise that in solving an equation, the variable is isolated, and understand that this will give the value for variable (unknown).
• Recognise that an equation is balanced around the equals symbol.
• Recognise that to isolate the variable, you have to ‘undo’ what has been done to it.
• Solve single step problems.
• Describe in words the process of finding the solution to a problem.
• Write equations, using whole numbers and variables, to represent problem situations.

Activity 1

1. Place in front of the students balance scales and plastic cubes.  Refer to the scenarios recorded in Session 2, Activity 3, Step 1 - in particular n + 5 = 17.
Ask students how they would represent this equation, using the equipment.
Ask: “What do we know about each side of this equation?” (They are both 17. We know this because of the = symbol).
Ask students to demonstrate, using the equipment, how the value of n is found.
Record the ‘process’ in an equation:
n + 5 – 5 = 17  5
Highlight these points:
• Remove 5 cubes from the left side, leaving n, and remove 5 cubes from the right side.
• 5 are removed from both sides to keep the scales in balance.
• An equation is like an object that balances.
• The original addition equation is solved by using the inverse operation of subtraction.
• + 5 – 5 = 0 : this is the additive inverse. It leaves us with n (+0) on the left side.
• We are ‘isolating n’ by undoing what has been done to it.
• n = 12.
2. Pose a further addition problem. eg. n + 10 = 22.
Have a student model the equation with equipment and record the process:
(n + 10 – 10 = 22 – 10 so n = 12)
3. Have students individually complete Attachment 3 (It’s all about Balance) Part A.
4. Agree as a class that to solve the two subtraction equations, the same amount was added to both sides of the equation.

Activity 2

1. Write 2n = 26 on the class modelling book.
Have students visualise how they would use the equipment to demonstrate the solution to this equation.
Have them share their ideas and, if necessary, demonstrate dividing both sides of the equation (both sets of cubes) by 2 and record this.
 Note the representation: 2n = 26 = 13 n = 13 2 2
2.  Write n = 13 on the class chart. 2
 Discuss n , highlighting that this is another way of saying half n, or half of n, is 13. 2
Have students visualise how they would use the scales and cubes to demonstrate the solution to this equation.
Have them share their ideas and, if necessary, demonstrate multiplying both sides of the equation (both sets of cubes) by 2 and record this.
 Note the representation: n x 2 = 13 x 2 n = 26 2 1
3. Have students individually complete Attachment 3 (It’s all about Balance) Part B.
4. Agree as a class that to solve the four equations, the inverse operation was used.
5. Review key, and important learning from today’s session.

Session 4

SLOs:

• Formally solve equations, which include variables and which represent problem situations involving fractional and decimal amounts.
• Estimate values for unknown amounts and explain reasoning.
• Recognise the calculator is a useful but ‘fallible’ tool and that errors are possible, and give explanations for these.

Activity 1

1. Explain that Kiwi School has been able to source cheaper supplies for their canteen lunches. Display Attachment 4 (Canteen Menu with changed prices), highlighting the new reduced prices.
Take time to review the summary of important learning from Session 3, Activity 2, Step 5 (above).
2. Make pencils, paper and calculators available.
Distribute a copy of Attachment 5 (Canteen order form: Thursday), Part A to each student. Discuss the task, highlighting the following:
• The value of the unknowns is to be found using equations in which each side is formally treated in the same way.
• The variables (a, b, n) may be for the price of the items, the number of items or for the total amount.
• Think carefully about the information given in each equation, and what fixed but unknown number the variable is representing.
Have the students individually complete the equations and order form for Thursday, then pair-share and fully discuss their results.
3. Gather as a class to discuss results and questions that may arise. Use the menu context to bring ‘reality’ to the discussion of each equation.
Highlight both common sense solutions and the formal way in which these equations can be solved.

Activity 2

1. Distribute Attachment 6 (Canteen order form: Friday).
Ensure calculators are made available.
Refer to the format of the Thursday order form. Identify the similarities.
2. Explain that the purpose of this task is to:
• estimate the value for the unknown using a common sense approach
• explain the errors in the completed order form for Friday.
Have students agree that sometimes we make errors when we enter calculations into a calculator. It is therefore very important that we have a sense of the size of the unknown value and that we do this by estimating, using our number sense as we look at the numbers involved.
3. Explore several examples before students discuss and complete this task in pairs, recording agreed results on their individual task sheets.
4. Have students pair-share their results, with each student amending their own form as appropriate.
5. Agree as a class how easily we can make or record calculator errors and how estimating amounts first can be very useful.

Activity 3

Conclude this session by revisiting key learning as noted on the class modelling book over four lessons. Have students reflect on and discuss their own learning.

Attachments