Unknowns in Number Sentences: a week of ready-to-teach maths

Five days of lessons for Year 6 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus loop cards, number strips with bracket and target cards, a mini-check and every answer.

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. Cut out the strips, bracket and target cards (Day 2) and the loop cards (Days 3 and 5) once.
4. Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and answers are all on the one page.

No maths background needed

This pack is written for the busy generalist teacher. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. The big idea of the week: an equation is a balance, and an unknown is found by undoing what was done to it. Each operation has an inverse that undoes it. When two things were done, they are undone in reverse order, like taking off shoes before socks. Brackets and the order of operations decide what was done first, and every answer can be checked by putting it back into the equation.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 50 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point if your timetable runs two shorter sessions.

Words to watch

The worksheets use a box, □, for the unknown; the board uses the letter x. Both mean the same: a number we do not know yet. The equals sign means “has the same value as”, not “the answer comes next”.

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 finds unknowns behind one operation; Day 2 fixes the order in which operations happen; Day 3 undoes two operations or brackets in reverse order; Day 4 finds unknowns with properties such as 25 × 4 = 100 instead of long calculation; and Day 5 checks answers and solves word problems. It builds on the Year 5 work of finding unknowns in sentences with one operation, and it prepares Year 7, where equations with letters are solved by the same balancing moves.

Materials for the week (one trip)

• From the classroom: scissors, pencils, this pack printed.
• For Day 1 (optional): a classroom balance or a coat hanger with two cups, and counters, to model an equation.
• Cut out once: the number strips, bracket cards and target cards, one set per pair, and the loop cards, one set per group of four.

Words for the word wall

equation, unknown, balance, inverse operation, undo, brackets, order of operations, substitute, check, commutative, associative, distributive

Dear families

This week in maths, Year 6 finds unknown numbers in number sentences such as 3 × (□ + 2) = 21. We undo what was done to the unknown, work in the right order (brackets first, then multiply and divide, then add and subtract), and check answers by putting them back in.

Try this at home

• Play “think of a number”: I double it and add 3 and get 19. What was my number? Swap roles.
• Ask: is 2 + 3 × 4 equal to 20 or 14? Who is right, and why?
• Use a shopping total: 3 drinks and a $4 sandwich cost $13. What does one drink cost?
• After solving, check together by putting the answer back into the sentence.

My number sentences this week

Fill one row a day. Write a number sentence with a box, your answer, and your check.

Printed from the free seegongsik Unknowns in Number Sentences teaching pack · seegongsik.com/au/y6/algebra/AC9M6A02/pack

Balance and undo

An equation says two sides have the same value, like a level balance. To find an unknown, undo what was done to it with the inverse operation: subtraction undoes addition, division undoes multiplication, and the other way round. Doing the same to both sides keeps the balance level.

We are learning to

• read an equation as a balance of two equal sides,
• name the inverse of each operation,
• find an unknown behind one operation of any kind.

Success criteria

• I can explain how to find □ in □ + 8 = 23 and in 4 × □ = 36.
• I can write an equation for a think-of-a-number puzzle and solve it.

You need

The worksheet, one per child, and a classroom balance with counters if you have one.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the inverse picture on the board.

Watch for these ideas

• Doing the operation instead of undoing it: □ + 8 = 23 does not give 31.
• Changing one side only, so the balance tips.
• Reading the equals sign as “the answer comes next”: 12 + □ = 30 does not give 42.

Answers

• Part A: □ + 5 = 12, so □ = 7; □ + 9 = 15, so □ = 6.
• Part B: □ + 8 = 23: take away 8, □ = 15; □ − 7 = 9: add 7, □ = 16; 4 × □ = 36: divide by 4, □ = 9; □ ÷ 5 = 6: multiply by 5, □ = 30; 12 + □ = 30: take 12 away from 30, □ = 18; □ + 2.5 = 7: take away 2.5, □ = 4.5; 56 ÷ □ = 8: 56 divided by 8, □ = 7.
• Part C: □ + 9 = 23, □ = 14; 6 × □ = 54, □ = 9; □ ÷ 4 = 12, □ = 48; 35 − □ = 18, □ = 17.

Balance and undo

A. Read the balance

Every bag holds the same number of marbles. Write the equation, then find how many are in the bag.

B. Undo it

C. Think of a number

Write each puzzle as an equation with a box, then solve it.

1. I think of a number and add 9. I get 23.   Equation: ____________________   □ = ______

2. I think of a number and multiply it by 6. I get 54.   Equation: ____________________   □ = ______

3. I think of a number and divide it by 4. I get 12.   Equation: ____________________   □ = ______

4. 35 take away my number is 18.   Equation: ____________________   □ = ______

Order of operations

An expression must mean the same to everyone, so there is an agreed order: brackets first, then multiplying and dividing from left to right, then adding and subtracting from left to right. Children work out expressions with and without brackets, judge number sentences, and place brackets to make sentences true.

We are learning to

• work out expressions in the agreed order of operations,
• explain how brackets change the value of an expression,
• place brackets to make a number sentence true.

Success criteria

• I can explain why 2 + 3 × 4 is 14, not 20.
• I can place brackets in 4 + 6 × 5 to make 50.

You need

The worksheet, one per child, and the number strips, bracket cards and target cards (cut-outs), one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the true-or-false sentences.

Watch for these ideas

• Working left to right whatever the operations: 2 + 3 × 4 is not 20.
• Doing all multiplying before dividing, or all adding before subtracting: each pair goes left to right.
• Ignoring brackets around an addition or a subtraction.

Answers

• Part A: 14 and 20; 14 and 2; 14 and 30; 26 and 5.
• Part B: true; false (14); false (8); true; true; false (5).
• Part C: (4 + 6) × 5 = 50; (20 − 8) ÷ 4 = 3; 3 × (8 − 2) × 4 = 72; 36 ÷ (4 + 5) = 4.

Order of operations

A. With and without brackets

B. True or false?

C. Put in brackets to make it true

Write each sentence again with one pair of brackets.

1. 4 + 6 × 5 = 50  →  ______________________________

2. 20 − 8 ÷ 4 = 3  →  ______________________________

3. 3 × 8 − 2 × 4 = 72  →  ______________________________

4. 36 ÷ 4 + 5 = 4  →  ______________________________

Two steps and brackets

When two things are done to an unknown, undo them in reverse order, the way shoes come off before socks. A flow diagram shows the order: forwards from the box to the answer, then backwards with each inverse. Brackets say what was done first, so they are undone last.

We are learning to

• show the order of the steps in an equation with a flow diagram,
• find an unknown by undoing two steps in reverse order,
• find an unknown inside brackets.

Success criteria

• I can find □ in 3 × □ + 2 = 17 and show the steps backwards.
• I can explain why 5 × (□ − 3) = 35 is undone with ÷ 5 first.

You need

The worksheet, one per child, and the loop cards (cut-outs), one set per group of four.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the balances in Part B.

Watch for these ideas

• Undoing in the forward order: for 3 × □ + 2 = 17, dividing by 3 first gives 17 ÷ 3.
• Undoing the operation inside the brackets first.
• Forgetting that halving is undone by doubling.

Answers

• Part A: 3 × □ + 2 = 17: − 2, then ÷ 3, □ = 5; 4 × □ − 7 = 25: + 7, then ÷ 4, □ = 8; □ ÷ 2 + 6 = 15: − 6, then × 2, □ = 18; 5 × (□ − 3) = 35: ÷ 5, then + 3, □ = 10; (□ + 4) × 2 = 30: ÷ 2, then − 4, □ = 11; (□ + 9) ÷ 3 = 7: × 3, then − 9, □ = 12.
• Part B: 2 × □ + 5 = 21, □ = 8; 3 × □ + 1 = 22, □ = 7.
• Part C: 4 × □ + 3 = 35, □ = 8; (□ + 5) × 3 = 36, □ = 7; □ ÷ 2 − 4 = 6, □ = 20; 6 × (□ − 3) = 42, □ = 10.

Two steps and brackets

A. Forwards, then backwards

Write the steps done to the box, then undo them in reverse order.

B. Bags on a balance

Every bag holds the same number of marbles. Write the equation and find how many are in a bag.

C. Think of a number

1. I multiply my number by 4 and add 3. I get 35.   Equation: ____________________   □ = ______

2. I add 5 to my number, then multiply by 3. I get 36.   Equation: ____________________   □ = ______

3. I halve my number and take away 4. I get 6.   Equation: ____________________   □ = ______

4. I take 3 from my number and multiply by 6. I get 42.   Equation: ____________________   □ = ______

Use the properties

Properties of numbers often find an unknown faster than calculating. Adding or multiplying can be done in any order and grouped either way, a product can be split into parts, and equal changes keep a sum or a difference the same. Children find unknowns this way and decide whether a sentence is always, sometimes or never true.

We are learning to

• use properties of numbers to find an unknown without long calculation,
• say which property they used,
• decide whether a sentence with a box is always, sometimes or never true.

Success criteria

• I can find □ in 25 × □ × 4 = 700 without working out 25 × □ first.
• I can explain why □ − 3 = 3 − □ is only sometimes true.

You need

The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the always-sometimes-never sentences.

Watch for these ideas

• Working out both sides in full when a property makes it quick.
• Swapping the numbers in a subtraction or a division as if the order did not matter.
• Changing both numbers of a sum the same way: 38 + 47 is not 40 + 49.

Answers

• Part A: 7 (any order: 25 × 4 = 100, so 100 × □ = 700); 4 (split a product: 14 is 10 + 4); 45 (equal sums: 38 went up 2, so 47 goes down 2); 4 (double one, halve the other); 7 (split a product: 99 is 100 − 1); 30 (equal differences: add 1 to both numbers); 1.4 (any order for adding); 3 (halve both numbers in a division).
• Part B: always; sometimes (only when □ = 3); never; always.
• Part C: any sentence a property solves quickly, such as 4 × □ × 25 = 900 (9).

Use the properties

A. Find the box the quick way

Do not work out both sides in full. Use a property, then say which one helped.

B. Always, sometimes or never true?

Try some numbers in the box. Circle one, and give an example.

C. Your own

Write a number sentence with a box that a property solves quickly. Give it to a partner to solve.

____________________________________________    □ = ______

Check and solve

Any answer can be checked by putting it back into the equation: if both sides come out the same, it is right. Children check given answers and correct the wrong ones, then write and solve equations for word problems, checking each one.

We are learning to

• check a solution by putting it back into the equation,
• write an equation for a word problem,
• solve it and answer the question in words.

Success criteria

• I can show that □ = 8 does not solve 4 × □ − 6 = 30, and find the right answer.
• I can write and solve an equation for the taxi problem.

You need

The worksheet, one per child, and the loop cards (cut-outs), one set per group of four.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the taxi problem.

Watch for these ideas

• Checking by solving again the same way, which repeats any slip.
• Putting the answer back into a changed line of working instead of the first equation.
• Answering a word problem with the box value alone, with no unit or sentence.

Answers

• Part A: 5 is right (21 = 21); 8 is wrong (it gives 26); the answer is 9; 12 is wrong (it gives 14); the answer is 14; 18 is right (10 = 10).
• Part B: 4 + 2 × □ = 26, so 11 km (check: 4 + 2 × 11 = 26); 3 × (□ + 6) = 45, so $9 (check: 3 × (9 + 6) = 45); □ ÷ 4 − 2 = 5, so 28 cm (check: 28 ÷ 4 − 2 = 5).

Check and solve

A. Is the answer right?

Put each answer back into its equation. If it is wrong, find the right answer.

B. Write it, solve it, check it

Loop cards

Cut out the sixteen cards once, one set for each group of four.

Loop race (Days 3 and 5)

1. Deal all sixteen cards among the group. The player with START reads the question at the bottom of the card.
2. Everyone works it out. The player whose card says “I have” that number reads it, then reads their own question.
3. Carry on until the question comes back to the START card. Time the whole loop.
4. On Day 5, play again and try to beat the time.

Teacher note: the loop in order is □ + 8 = 15 (7), 4 × □ = 36 (9), 3 × □ + 2 = 17 (5), (□ + 4) × 2 = 30 (11), □ − 6 = 9 (15), □ ÷ 5 = 6 (30), 2 × (□ − 3) = 18 (12), 48 ÷ □ = 8 (6), 5 × □ − 4 = 36 (8), □ ÷ 2 + 6 = 16 (20), 3 × (□ + 1) = 42 (13), 100 − □ = 82 (18), (□ − 2) ÷ 3 = 5 (17), 6 × □ + 5 = 29 (4), 25 + □ × 2 = 45 (10), 9 × (□ − 1) = 18 (3), and back to the start.

Number strips, brackets and targets

Cut out the three strips, the bracket cards and the target cards once, one set for each pair.

Bracket it (Day 2)

1. Lay the three strips face up. Shuffle the target cards face down.
2. Turn over a target card. Both players use bracket cards on its strip to make the target. No brackets at all can be right too.
3. Work it out aloud to check. The first correct placement wins the target card.

Teacher note: 17 is 6 + 4 × 3 − 1; 29 is (6 + 4) × 3 − 1; 14 is 6 + 4 × (3 − 1); 20 is (6 + 4) × (3 − 1); 10 is 20 − 4 × 3 + 2; 50 is (20 − 4) × 3 + 2; 0 is 20 − 4 × (3 + 2); 80 is (20 − 4) × (3 + 2); 6 is 20 − (4 × 3 + 2); 24 is 5 × 6 − 2 × 3; 60 is 5 × (6 − 2) × 3; 0 is 5 × (6 − 2 × 3); 84 is (5 × 6 − 2) × 3.

What we know: Unknowns in Number Sentences

Work on your own. Show your thinking if you can.

1. □ + 17 = 42   □ = ______
2. 7 × □ = 63   □ = ______
3. Work out 3 + 5 × 4 = ______ and (3 + 5) × 4 = ______
4. Put in brackets to make it true: 2 × 5 + 3 = 16
5. 4 × □ − 5 = 31   □ = ______
6. 2 × (□ + 7) = 30   □ = ______
7. Use a property, not a long calculation: 46 + 38 = 44 + □   □ = ______   How did you know? ____________________
8. Is □ = 6 the answer to 3 × (□ − 2) = 15? Show your check. If not, find the answer. ______________________________

Answers and marking guide

Answers

1. 25 (take away 17).
2. 9 (divide by 7).
3. 23 and 32.
4. 2 × (5 + 3) = 16.
5. 9 (add 5, then divide by 4).
6. 8 (divide by 2, then take away 7).
7. 40: 46 went down 2 to make 44, so 38 goes up 2.
8. No: it gives 12, not 15. □ = 7.

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and can check each answer by putting it back.

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: balance and undo, the order of operations, two steps and brackets, the properties, and checking and solving.