Formulating algebraic expressions: a week of ready-to-teach maths

Five lessons for Year 7 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus operation word cards, “I have, who has” loop cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets: operation word cards, one per pair (Day 1), and one set of loop cards for the class (Day 5).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

An expression is a recipe written in symbols. It is built from variables (letters that can take any value), constants (fixed numbers), operations and brackets. “A number, doubled, then add 3” becomes 2n + 3, one phrase at a time. Brackets change the meaning: 2(n + 3) doubles the whole of n + 3, so it is always 3 more than 2n + 3. A few phrases reverse the order you read them: 5 less than n is n − 5, not 5 − n. Testing an expression with a real number catches most slips.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write expressions

Write a number before a letter for multiplication, 3n, and leave out the × sign. Write division with ÷ or as a fraction, n ÷ 4 or n/4. A bracket next to a number means multiply the whole bracket: 2(n + 3). An expression has no equals sign; it is a value waiting for a number. Calculators are not needed this week.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 translates words into symbols; Day 2 names the parts of an expression; Day 3 adds brackets; Day 4 deals with the phrases that reverse the order; and Day 5 writes expressions for real situations and growing patterns. It builds on substituting into formulas (AC9M7A01) and prepares the equations of AC9M7A03, where an expression is set equal to a value and solved.

Materials for the week (one trip)

• From the classroom: pencils, scissors, small envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 16 operation word cards (Day 1).
• Cut-out sheet 2, one for the class: 12 “I have, who has” loop cards (Day 5).
• Matchsticks or toothpicks for the growing pattern on Day 5, if you have them; drawings work too.

Words for the word wall

expression, variable, constant, coefficient, term, operation, bracket, sum, difference, product, quotient, more than, less than, increased by, decreased by, double, triple, halve, evaluate, pattern

Dear families

This week in maths, Year 7 is writing algebraic expressions: turning rules in words, such as “double a number, then add 3”, into symbols such as 2n + 3, and checking them with real numbers.

Try this at home

• Say a rule in words, such as “triple a number and take away 1”. Can your child write it in symbols? (3n − 1.)
• Ask: what is 5 less than 12? Is that 12 − 5 or 5 − 12?
• A movie ticket costs $15 and popcorn $6. Write the cost for t tickets and p popcorns. (15t + 6p.)
• Ask: is 2(n + 4) the same as 2n + 4? Try n = 1.

Expressions this week

Fill one row a day with a rule in words, its expression, and its value for one number.

Printed from the free seegongsik Formulating algebraic expressions teaching pack · seegongsik.com/au/y7/algebra/AC9M7A02/pack

Words into symbols

Operation words map to symbols: more than means add, less than means subtract, product means multiply and quotient means divide. Students sort the words, then translate worded rules into expressions one phrase at a time, and back into words.

We are learning to

• match operation words to the four operations,
• translate a worded rule into an expression, one phrase at a time,
• describe an expression in words.

Success criteria

• I can write “halve a number, then add 6” as n ÷ 2 + 6.
• I can say an expression such as 9n − 2 in words.

You need

The operation word cards from cut-out sheet 1 (one set per pair), the worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the word sort; start the second with the board.

Watch for these ideas

• Reversing division: a number divided by 3 is n ÷ 3, not 3 ÷ n.
• Writing 4 × n with a letter x for times. Write 4n.
• Translating less than in reading order. Day 4 comes back to it.

Answers

• Part A: n + 9, 4n, n ÷ 3, n − 7, 2n + 5, 10 − n, 6n, 3n − 1.
• Part B, for example: x + 8, a number plus 8; 5x, 5 times a number; x − 2, 2 less than a number; 4x + 3, four times a number, then add 3.
• Part C: 2c + 4p, 4.5t, m ÷ 60.
• Part D: for example, “twice a number, minus 1” and “1 less than double a number”. n − 20 = −10 and 20 − n = 10, so n less than 20 is larger.
• Word cards: add: sum, more than, increased by, total; subtract: difference, less than, decreased by, fewer than; multiply: product, times, double, triple; divide: quotient, divided by, shared equally, half of.

Words into symbols

Let n stand for “a number” unless the question gives a letter.

A. Translate into an expression

B. Say it in words

1. x + 8 ________________________________
2. 5x ________________________________
3. x − 2 ________________________________
4. 4x + 3 ________________________________

C. Choose the letters, then write the expression

1. the number of legs on c chickens and p pigs: ________________
2. the cost of t tram rides at $4.50 each: ________________
3. a number of minutes m, written in hours: ________________

D. Stretch

Write two different phrases in words for 2n − 1. ________________________________

When n = 10, which is larger: “20 less than n” or “n less than 20”? Show both values. ________________

Parts of an expression

An expression is built from named parts: in 3n + 7, 3 is the coefficient, n the variable and 7 the constant. A term is a part joined to the rest by + or −. Naming the parts shows what to change when a rule changes.

We are learning to

• name the coefficients, variables, constant and terms of an expression,
• write repeated addition as a product, such as n + n + n = 3n,
• evaluate an expression for given values.

Success criteria

• I can name every part of 8m + 1.
• I can evaluate 2a + 5 when a = 6.

You need

The worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the boxes; start the second with Part A.

Watch for these ideas

• Reading 3n as a two-digit number. 3n means 3 × n.
• Joining unlike parts: a + b is not 2ab, and 2a + 3 is not 5a.
• Missing that b on its own has coefficient 1.

Answers

• Part A: 5n + 2: coefficient 5, variable n, constant 2, 2 terms; 6 + 2y: coefficient 2, variable y, constant 6, 2 terms; 4a + b + 9: coefficient 4 and 1, variable a and b, constant 9, 3 terms; 10x: coefficient 10, variable x, constant none, 1 term.
• Part B: n + n + n + n = 4n, a + a + b + b + b = 2a + 3b, x + 5 + x = 2x + 5; 3y = y + y + y.
• Part C: 3n + 7 = 22, 4a + b + 9 = 20, 6 + 2y = 24, 10x = 5.
• Part D: 6t + 1 (or 1 + 6t). For example 5n, n + 16 and 4n + 4.

Parts of an expression

A. Name the parts

B. Write it shorter, or longer

C. Evaluate

D. Stretch

Write an expression with coefficient 6, constant 1 and variable t. ________________

Write three different expressions that are each equal to 20 when n = 4. ________________________________

Brackets

Brackets group the parts that are handled together. 2(n + 3) adds 3 first, then doubles; 2n + 3 doubles first, then adds 3, so it is always 3 smaller. Students write rules that need brackets and test expressions with numbers.

We are learning to

• explain what a bracket groups,
• write a worded rule that needs brackets,
• test two expressions with the same number.

Success criteria

• I can show that 3(n + 1) and 3n + 1 differ when n = 2.
• I can write “triple the sum of a number and 4” as 3(n + 4).

You need

The worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the rules; start the second with Part A.

Watch for these ideas

• Multiplying only the first term: 2(n + 3) is not 2n + 3.
• Adding brackets where none are needed: 2n + 3 already means double, then add.
• Thinking brackets are decoration. Test both forms with a number.

Answers

• Part A (n = 4): 2(n + 3) = 14, 2n + 3 = 11; 5(n − 1) = 15, 5n − 1 = 19; (n + 6) ÷ 2 = 5, n + 6 ÷ 2 = 7.
• Part B: 2(n + 5), 3(n − 2), (n + 4) ÷ 2, 6(n + 1).
• Part C: 2(x + 3), 4(k + 1), 4(d + s).
• Part D: yes; n = 1 gives 9 both ways, n = 2 gives 12 both ways, n = 10 gives 36 both ways. A rectangle 3 by (n + 2) splits into 3 by n and 3 by 2, so 3n + 6.

Brackets

A. With and without brackets

Work out both expressions when n = 4.

B. Write the expression

1. double the sum of a number and 5: ________________
2. three times the difference of a number and 2: ________________
3. the sum of a number and 4, divided by 2: ________________
4. add 1 to a number, then multiply by 6: ________________

C. Brackets in shapes and costs

1. A rectangle is (x + 3) cm long and 2 cm wide. Write its area. ________________
2. A square has sides (k + 1) cm long. Write its perimeter. ________________
3. Four friends each buy a drink costing d dollars and a snack costing s dollars. Write the total cost. ________________

D. Stretch

Is 3(n + 2) equal to 3n + 6? Test n = 1, 2, 10, then explain with a drawing of a rectangle 3 by (n + 2). ________________________________

Order matters

Most phrases translate in the order you read them, but less than, fewer than and subtract from reverse it: 5 less than n is n − 5. Subtraction and division cannot be swapped, so testing with a real number settles every case.

We are learning to

• translate less than, subtract from and divided by in the right order,
• test a translation with a real number,
• correct a reversed translation.

Success criteria

• I can explain why 4 less than n is n − 4, using n = 10.
• I can tell n ÷ 2 from 2 ÷ n with a number.

You need

The worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the pairs; start the second with Part A.

Watch for these ideas

• Writing 5 − n for 5 less than n, in reading order.
• Thinking subtraction or division can be swapped like addition: 3 − n and n − 3 differ.
• Trusting a translation without testing it with one number.

Answers

• Part A and B (n = 12): n − 5 = 7; 5 − n = −7; n ÷ 4 = 3; 4 ÷ n = 1/3; n − 3 = 9; 3 − n = −9; n − 8 = 4; 8 − n = −4.
• Part C: Sam: 8 − n gives −12 at n = 20, but it should be 12; write n − 8. Jo: 5 ÷ n gives 1/6 at n = 30, but it should be 6; write n ÷ 5.
• Part D: n = 5, where both are 0. (20 − n) ÷ 2.

Order matters

A and B. Translate, then test with n = 12

C. Check and fix

1. Sam wrote “8 less than a number” as 8 − n. Test it with n = 20. Is it right? If not, fix it. ________________________
2. Jo wrote “a number shared equally among 5” as 5 ÷ n. Test it with n = 30. Is it right? If not, fix it. ________________________

D. Stretch

For which value of n do n − 5 and 5 − n give the same answer? ________

Write “a number subtracted from 20, then halved” as an expression. ________________

Expressions from situations

Real situations become expressions: a fixed charge plus a rate, a perimeter from labelled sides, a growing pattern. Students play the loop game, then choose letters and write expressions as simply as they can.

We are learning to

• write an expression for a situation and say what each letter stands for,
• write an expression for a growing pattern,
• collect like terms to write an expression simply.

Success criteria

• I can write the cost of t tickets at $15 each with a $4 booking fee.
• I can find the rule for a matchstick pattern and use it for a large shape.

You need

The loop cards from cut-out sheet 2 (one set for the class), matchsticks if you have them, the worksheet, the self-check quiz.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Charging the fixed part every hour: the cost is 90 + 75h, not 165h.
• Combining unlike terms, such as x + 2 into 2x.
• Drawing every shape of a pattern. Look for what is added each time.

Answers with working

1. 90 + 75h dollars (the 90 is paid once).
2. x + (x + 2) + 2x = 4x + 2 centimetres.
3. 4, 7, 10: 3 more each time, so 3n + 1; for 20 squares, 3 × 20 + 1 = 61.
4. brother m + 3, aunt 4m; sum m + (m + 3) + 4m = 6m + 3.
5. 650n + 40 dollars.
6. 18a + 12c; 18 × 2 + 12 × 3 = $72.
7. (3n + 4) ÷ 2 dollars each; the bracket keeps the whole cost together.
8. length 2w + 3; perimeter 2(2w + 3) + 2w = 6w + 6 centimetres.

Loop order: 3n + 1 → n − 6 → 2(n + 4) → n ÷ 5 → 6 − n → 4n → n + 10 → 2n + 4 → 5 ÷ n → 3(n − 1) → 10n → 3n − 1 → back to the start.

Expressions from situations

Say what each letter stands for, then write the expression as simply as you can.

Problems (7 and 8 are the stretch)

1. A plumber charges $90 to come out, plus $75 an hour. Write an expression for the cost of a job that takes h hours.
2. A triangle has sides of x, x + 2 and 2x centimetres. Write its perimeter as simply as you can.
3. The matchstick pattern below grows by one square each time. Write an expression for the matches in n squares, then find the number for 20 squares.
   1 square

   2 squares

   3 squares
4. Mia is m years old. Mia’s brother is 3 years older, and their aunt is 4 times Mia’s age. Write each age, then the sum of all three ages as simply as you can.
5. A school buys n laptops at $650 each, plus a $40 delivery fee. Write the total cost.
6. Cinema tickets are $18 for adults and $12 for children. Write the cost for a adults and c children, then find it for 2 adults and 3 children.
7. Stretch. 3 notebooks at n dollars each and a $4 pen are paid for together, and the cost is shared equally by 2 friends. Write each friend’s share.
8. Stretch. A rectangle’s length is 3 cm more than twice its width, w cm. Write its perimeter as simply as you can.

Operation word cards

Cut out the 16 cards. Day 1: sort them into four piles: add, subtract, multiply and divide.

“I have, who has” loop cards

Cut out the 12 cards and hand them round the class. Day 5: the student with the start card reads it aloud; whoever holds the answer reads theirs next, until the loop comes back to the start.

What we know: Algebraic expressions

Work on your own, without a calculator. Use n for “a number”.

1. Write “a number increased by 12”. ________
2. Write “9 less than a number”. ________
3. In 6p − 4, name the coefficient, the variable and the constant. ________________________
4. Write “three times the sum of a number and 7”. ________
5. Work out 2(x + 5) and 2x + 5 when x = 6. ________________
6. Work out n ÷ 4 and 4 ÷ n when n = 8. ________________
7. Pens cost $2 and notebooks $5. Write the cost of p pens and b notebooks. ________
8. A row of matchstick triangles uses 3, 5, 7 matches for 1, 2 and 3 triangles. Write the number of matches for n triangles. ________

Answers and marking guide

Answers with working

1. n + 12.
2. n − 9 (test: 9 less than 20 is 11, and 20 − 9 = 11).
3. Coefficient 6, variable p, constant 4 (subtracted).
4. 3(n + 7).
5. 2(6 + 5) = 22 and 2 × 6 + 5 = 17.
6. 8 ÷ 4 = 2 and 4 ÷ 8 = 1/2.
7. 2p + 5b dollars.
8. 2 more matches each time, so 2n + 1; check: 2 × 3 + 1 = 7.

A quick three-level guide

Eight questions, four ideas. A student at standard writes most expressions correctly and tests an expression with a number when the order is in doubt.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: words into symbols, parts, brackets, order, and situations.