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

Five days of lessons for Year 4 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus cut-out true-or-false cards and box cards with their numbers, 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 two card sheets once; the cards are reused across the week.
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 is that the equals sign means the two sides are the same, like a balanced scale, not that the answer comes next. Then a missing number is something to reason about: what keeps the two sides level?

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.

The box, the whole and the parts

A box (□) stands for one missing number. In 8 + □ = 15, the 15 is the whole and 8 and the box are its parts. In □ − 5 = 11, the box is the whole. To find a whole, add the parts; to find a part, take the other part from the whole. Children check every answer by putting it in the box and reading the sentence.

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 fixes the meaning of the equals sign, because everything else depends on it. Days 2 and 3 find a box by reasoning about wholes and parts and by undoing. Day 4 puts a calculation on both sides and uses the properties of numbers, such as order not mattering in addition, to find a box without working everything out. Day 5 brings it together with stories. It builds on Year 3’s work on addition and subtraction undoing each other (AC9M3A01).

Materials for the week (one trip)

• From the classroom: scissors, pencils, this pack printed, and a pan balance or coat-hanger balance if you have one.
• Optional: interlocking cubes, so children can build both sides of a sentence.
• Cut out once, use all week: the true-or-false cards and the box cards in this pack.

Words for the word wall

equals, the same as, balance, true, false, unknown, box, whole, part, inverse, undo, check, both sides

Dear families

This week in maths, Year 4 learns that the equals sign means “is the same as”, and uses that to find missing numbers in sentences such as 8 + □ = 15.

Try this at home

• Ask true or false: is 12 = 5 + 7 true? Is 8 + 2 = 10 + 3 true? Talk about why.
• Hide some pegs: I had 15 pegs, I hid some, and 8 are left. How many did I hide?
• Make both sides match: 6 + 9 = 5 + what? Can your child do it without adding 6 and 9?
• Turn a shopping moment into a sentence: I had $20, spent some, and have $7 left.

My number sentences this week

Fill one row a day. Write a sentence with a box, the number in the box, and how you checked it.

Printed from the free seegongsik Finding Unknowns teaching pack · seegongsik.com/au/y4/algebra/AC9M4A01/pack

The equals sign means balance

Many children read the equals sign as “the answer comes next”, so 12 = 5 + 7 looks backwards to them and 8 + 2 = 10 + 3 looks fine. Today it means “is the same as”: a sentence is true when both sides have the same value, like a level balance, wherever the calculation sits.

We are learning to

• read the equals sign as “is the same as”,
• decide whether a number sentence is true by working out each side,
• make a false sentence true by changing one number.

Success criteria

• I can say whether a number sentence is true and explain why.
• I can write a true sentence with the answer first, or a calculation on both sides.

You need

A pan balance and cubes if you have them. The true-or-false cards (cut-out sheet 1), one set per pair. The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Reading = as “the answer comes next”, and so calling 12 = 5 + 7 false.
• Running on: 8 + 2 = 10 + 3 = 13 writes 10 = 13 in the middle, which is false.
• Thinking 15 = 15 is not a real sentence. It is true: both sides are the same.

Answers

• Part A: 7 + 6 = 13 true; 12 = 5 + 7 true; 9 + 4 = 14 false (13 and 14); 15 = 15 true; 8 + 2 = 10 + 3 false (10 and 13); 6 + 6 = 10 + 2 true; 18 − 9 = 9 true; 20 − 5 = 10 + 4 false (15 and 14).
• Part B: changes vary, for example 9 + 4 = 13, 8 + 2 = 10 + 0 and 20 − 5 = 10 + 5. Part C: any true sentence with the answer first, such as 12 = 5 + 7. Part D: 12 cubes in each pan, so it is level. Card sort: 7 true and 5 false.

The equals sign means balance

A. True or false? Work out each side.

B. Change one number in each false sentence to make it true

C. Sam says the equals sign means “the answer comes next”. Write a true sentence that shows he is wrong.

Find the unknown

A box stands for a missing number. Children find it by asking which number is the whole and which are the parts. In 8 + □ = 15 the whole is 15, so the box is the part left when 8 is taken away: 7. In □ − 5 = 11 the box is the whole, made of 5 and 11: 16. Every answer is checked by putting it in the box.

We are learning to

• name the whole and the parts in an addition or subtraction sentence,
• find the box: add the parts for a whole, subtract for a part,
• check by reading the sentence with the number in the box.

Success criteria

• I can find the number in a box wherever the box is.
• I can say whether the box is the whole or a part.

You need

15 cubes and a cup. The box cards and number cards (cut-out sheet 2), one set per pair. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the first four rows of Part A; start Session B with □ + 25 = 40.

Watch for these ideas

• Adding every number in sight: 8 + 15 = 23 for 8 + □ = 15. The 15 is the whole already.
• With the box first in a subtraction, taking away: 11 − 5 = 6 for □ − 5 = 11. The box is the whole, so add.
• Skipping the check. Reading the sentence with the number in the box catches both slips.

Answers

• Part A: 8 + □ = 15: part, 15 − 8 = 7; □ + 6 = 13: part, 13 − 6 = 7; 20 − □ = 12: part, 20 − 12 = 8; □ − 5 = 11: whole, 11 + 5 = 16; □ + 25 = 40: part, 40 − 25 = 15; 34 − □ = 20: part, 34 − 20 = 14; □ − 18 = 30: whole, 30 + 18 = 48; 47 + □ = 60: part, 60 − 47 = 13.
• Part B: sentences vary; in a whole sentence the box is the total of an addition or the start of a subtraction. Card match: 3 of the 12 box cards have the box as the whole.

Find the unknown

A. Is the box the whole or a part? Find it, then check.

B. Write two box sentences of your own

One where the box is the whole:

One where the box is a part:

Undo to find it

Adding and subtracting undo each other. To free the box in □ + 9 = 16, undo the adding of 9: 16 − 9 = 7. To free it in □ − 7 = 8, undo the taking away of 7: 8 + 7 = 15. One kind needs care. In 18 − □ = 7 the box is what was taken away, so it is 18 − 7 = 11, not 18 + 7.

We are learning to

• say what was done to the box, and do the opposite,
• choose add or subtract to find the box,
• explain the case where the box itself is taken away.

Success criteria

• I can choose the inverse operation and find the box.
• I can check my answer by doing the sentence forwards.

You need

The box cards (cut-out sheet 2), one set per pair. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the first four rows of Part A; start Session B with □ − 26 = 14.

Watch for these ideas

• Undoing with the same operation: 16 + 9 for □ + 9 = 16.
• Adding when the box is taken away: 18 + 7 = 25 for 18 − □ = 7. The box is a part, so subtract.
• Not checking forwards. 25 in 18 − □ = 7 gives 18 − 25, which cannot leave 7.

Answers

• Part A: □ + 9 = 16: subtract, 16 − 9 = 7; □ − 7 = 8: add, 8 + 7 = 15; 4 + □ = 13: subtract, 13 − 4 = 9; 18 − □ = 7: subtract, 18 − 7 = 11; □ − 26 = 14: add, 14 + 26 = 40; 52 − □ = 30: subtract, 52 − 30 = 22; □ + 35 = 61: subtract, 61 − 35 = 26; □ − 19 = 45: add, 45 + 19 = 64.
• Part B: in □ − 7 = 8 the box is the whole and 7 was taken from it, so put the parts back together: 8 + 7. In 18 − □ = 7 the box is a part of 18, so take the other part away: 18 − 7.

Undo to find it

A. Add or subtract? Find the box, then check forwards.

B. Why does □ − 7 = 8 need adding, but 18 − □ = 7 needs subtracting?

Both sides balance

Now both sides of the equals sign hold a calculation, and the box is in one of them: 4 + 9 = 7 + □. One way is to work out the known side, 13, then find the box, 6. Another uses the properties of numbers: in 8 + 7 = 10 + □, 10 is 2 more than 8, so the box is 2 less than 7, which is 5. That second way saves work when the numbers are big.

We are learning to

• find a box in a sentence with a calculation on each side,
• use the properties of numbers: order does not change a sum, and if one number goes up the other must come down by the same amount,
• explain why the two sides still balance.

Success criteria

• I can find a box when both sides have a calculation.
• I can find some boxes without working out either side, and explain how.

You need

The true-or-false cards from Day 1. Cubes if you have them. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with 27 + 15 = 28 + □.

Watch for these ideas

• Writing the left side’s total in the box: 13 for 4 + 9 = 7 + □. Then the right side would be 20.
• Adding everything in sight: 4 + 9 + 7 = 20.
• Shifting both numbers the same way: 27 + 15 = 28 + 16 is out of balance by 2.

Answers

• Part A: 4 + 9 = 7 + □: both sides 13, box 6; 8 + 7 = 10 + □: both sides 15, box 5; 6 + 9 = 9 + □: both sides 15, box 6; 15 − 3 = 10 + □: both sides 12, box 2; 13 + 5 = □ + 6: both sides 18, box 12; 20 − 8 = 14 − □: both sides 12, box 2.
• Part B: 27 + 15 = 28 + □: 28 is 1 more than 27, so the box is 1 less than 15, which is 14; 46 + 38 = 50 + □: 50 is 4 more than 46, so the box is 4 less than 38, which is 34; 35 + 19 = 34 + □: 34 is 1 less than 35, so the box is 1 more than 19, which is 20. Part C: Kai wrote the total of the left side; the box is 6.

Both sides balance

A. Work out the known side, then find the box

B. Find the box without working out either side. How do you know?

C. Kai says the box in 4 + 9 = 7 + □ is 13. What did Kai do? What is the box?

Keep the balance

Many different sentences balance at the same total. Children find the boxes in sentences that all equal 20, then write their own that equal 25. Then they turn stories into box sentences: Jack had some money, spent $9 and has $16 left is □ − 9 = 16, and the box is the money he started with.

We are learning to

• find boxes in sentences that balance at the same total, including with the total first,
• write a story as a number sentence with a box,
• choose the inverse and answer the story in words.

Success criteria

• I can write a number sentence with a box for a story, and solve it.
• I can write different sentences that balance at the same total.

You need

The box cards and number cards (cut-out sheet 2) for a last match. The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Writing the working instead of the story: 24 − 15 = □ finds the box, but the story says 24 − □ = 15.
• Calling 20 = 9 + □ backwards. It is a true sentence once the box is 11.
• Answering a story with a bare number. Jack had $25 is the answer; 25 is not.

Answers

• Part A: 12 + □ = 20, box 8; □ + 15 = 20, box 5; 30 − □ = 20, box 10; □ − 6 = 20, box 26; 20 = 9 + □, box 11; 20 = □ − 4, box 24.
• Part B: 24 − □ = 15, box 9: 9 stickers; □ − 9 = 16, box 25: $25; 18 + □ = 31, box 13: 13 people. Part C: any four sentences that balance at 25, such as 25 = 30 − 5.

Keep the balance

A. Every sentence balances at 20. Find each box.

B. Write each story as a sentence with a box. Solve it and answer the story.

C. Write four different box sentences that balance at 25

Check each one by putting your box number in and reading it.

True-or-false cards

Cut out the two headers and the twelve sentence cards, one set per pair. Partners lay out TRUE and FALSE, then sort each sentence under one of them and explain it (Day 1). On Day 4, turn a false card into a true box sentence.

Headers

Sentence cards

Teacher note: 7 cards are true and 5 are false. Some put the answer first, and some have a calculation on both sides.

Box cards and number cards

Cut out the twelve box cards and the twelve number cards, one set per pair. Partners match each box card to the number that goes in its box, and check by reading the sentence with the number in place.

Box cards

Number cards

Teacher note: every box has a different number, so each card has exactly one partner.

What we know: Finding Unknowns in Number Sentences

Find each box, then check by reading the sentence with your number in it.

1. True or false?   7 + 5 = 12 ____    14 = 6 + 9 ____    8 + 3 = 5 + 6 ____
2. 9 + □ = 17   Box = ______
3. □ − 8 = 6   Box = ______
4. 25 − □ = 18   Box = ______
5. □ + 13 = 30   Box = ______
6. 6 + 9 = 5 + □   Box = ______   Can you do it without adding 6 and 9?
7. To find the box in □ − 12 = 20, do you add or subtract? Why? __________________   Box = ______
8. Lena had some money. Gran gave her $12 and now she has $30. Write a sentence with a box, then answer: ______________________

Answers and marking guide

Answers

1. true; false (14 and 15); true.
2. 17 − 9 = 8.
3. 6 + 8 = 14 (the box is the whole, so add).
4. 25 − 18 = 7 (the box is a part, so subtract).
5. 30 − 13 = 17.
6. 10: 5 is 1 less than 6, so the box is 1 more than 9.
7. Add: the box is the whole and 12 was taken from it, so 20 + 12 = 32.
8. □ + 12 = 30; 30 − 12 = 18, so Lena had $18 before Gran gave her the money.

A quick three-level guide

Eight questions, three ideas. A child at standard reads the equals sign as balance, finds a box wherever it sits, and can say whether they added or subtracted and why.

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: the equals sign, finding the box, undoing, both sides, and writing sentences.