What fills the blank in 13+4=square+6?

11. Both complete sides must equal 17.

Which is greater: 120÷3 or 35+25?

35+25. Its value is 60, while the other gives 40.

How does 113-25 compare with 112-24?

They are equal. The extra starting amount matches the extra amount taken away.

Subtract the bracketed sum of fifteen and fifty-six from one hundred. What comes first?

15 plus 56. The amount subtracted includes both costs.

In two minus ten plus four times six, which complete term contains multiplication?

4×6. The product contributes one value to the sum.

When rearranging eighty-three minus fourteen as a sum, what must move with fourteen?

The minus sign. The term is negative, and moving it must preserve its value.

Start with two hundred minus the bracketed sum of forty and three. After removing brackets, is three added or subtracted?

Subtract it too. The whole sum is subtracted, including the 3.

Two people each buy items costing forty-three and twenty-four rupees. Which two products, left uncalculated, add to their total?

2×43+2×24. Both costs occur twice.

Teach Arithmetic Expressions in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather six spoons, twelve chickpeas, eight raisins, ten buttons, a plate, paper and pencil.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

Shopping totals combine item costs into one amount.
Change calculations subtract everything spent from the money handed over.
Equal groups let us count objects without counting each separately.

What to teach

How to teach it

Expressions and their values

Six steel spoons arranged in groups of two and four.

1. Simple Expressions

A written calculation whose result is its value.

Show: Arrange six spoons in groups of two and four. Write their addition. After the prediction, regroup into threes. Write that addition and join both expressions with equals.
Ask: Will regrouping these spoons change their total?
Say: Both expressions have value six, so equals can join them.
Used in: Different combinations of coins can pay the same amount.

If your child says…“Equals means the answer comes next.”

Ask: Could either spoon grouping go on the left of equals?
What’s true: Equals connects complete sides with matching values.

Number line showing five minus four ending at one, two plus one at three.

2. Comparing Expressions

Compare the results of calculations.

Show: Find five minus four in the diagram. Follow its arrow, then follow the arrow for two plus one.
Ask: Does starting at five make five minus four greater than two plus one?
Say: Two plus one is greater. It ends at three, farther right than one on the number line.
Used in: Comparing final prices helps shoppers find the cheaper purchase.

If your child says…“The expression with the biggest number must win.”

Ask: Which arrow lands farther right on the number line?
What’s true: The numbers and operations together determine each expression’s value.

Rows of six and four chickpeas, with two spare chickpeas beside them.

3. Comparing Expressions

Track changes to compare results without calculating both.

Show: Arrange six chickpeas above four, matching vertical pairs. Write “six minus four”. After the prediction, add one to each row and write “seven minus five”.
Ask: Will adding one chickpea to each row change the difference?
Say: The new pair leaves the unmatched chickpeas unchanged, so the differences are equal.
Used in: Comparing price changes can show which shopping total rises more without recalculating everything.

If your child says…“Both rows gain one, so the difference grows by two.”

Ask: Does the added pair leave any extra chickpeas unmatched?
What’s true: Equal increases in both amounts leave their difference unchanged.

Reading and rearranging expressions

$(2+3)\times2=(2+3)+(2+3)$

$(2+3)+(2+3)=5+5$

4. Brackets in Expressions

Finish the enclosed calculation before using its result.

Show: Write this line. Say, “Add two and three, then double their sum.”
Ask: What result comes next when we double the whole sum?
Say: The result is ten. The brackets make us add two and three before doubling.
Used in: Brackets keep all spending together when calculating change.

If your child says…“I can leave the brackets until last.”

Ask: Are we doubling just three, or the sum of two and three?
What’s true: The bracketed group must be ready before its value is used outside.

$7-2\times3=7+(-2\times3)$

$7+(-2\times3)=7+(-6)$

5. Terms in Expressions

The complete signed parts added to form an expression.

Show: Say, “Subtracting means adding the opposite.” Write this line. Underline seven and the negative product separately.
Ask: After finishing each term, what value goes on the next line?
Say: Negative two times three gives negative six; adding seven gives one.
Used in: Bills keep each quantity-times-price calculation together, making item totals easier to check.

If your child says…“Two and three are separate terms.”

Ask: Are we adding two and three separately, or using their product?
What’s true: A multiplication supplies one value, so its factors stay together.

$3+(-2)+2=3+2+(-2)$

$3+2+(-2)=3+(2+(-2))$

$3+(2+(-2))=3+0$

6. Swapping and Grouping

Signed terms can change places or groups without changing the sum.

Show: Say, “Move each term with its sign, then group the opposite terms.” Write this line.
Ask: What remains on the next line?
Say: Group two with negative two: they add to zero, leaving three.
Used in: A forgotten expense can be added to an existing total without restarting the addition.

If your child says…“I can move the number and leave its minus behind.”

Ask: If negative two became positive two, would the pair still cancel?
What’s true: A term keeps its sign when you move it.

Removing brackets

Three raisins on a steel plate and five beside it.

7. Removing Brackets—I

Subtracting a bracketed group changes every term’s sign, including the first.

Show: Place eight raisins on a plate. Write “eight minus the difference of five and two”. Remove five. After the prediction, return two and write $8-(5-2)=8-5+2$ .
Ask: Should we return or remove two to take away two fewer than five?
Say: Taking five removes two too many, so return two.
Used in: Subtracting the full price leaves too little savings. Adding back the discount gives the correct amount.

If your child says…“I change only the first sign after the minus.”

Ask: If you removed two more, would you have taken away less than five?
What’s true: Subtracting a bracketed group reverses every internal sign when the brackets are removed.

Two rows of buttons, each split into groups of three and two.

8. Removing Brackets—II

Multiply each part of the bracketed group.

Show: Arrange ten buttons in two rows of five. Slide the last two in each row rightwards. After the prediction, write $2\times(3+2)=2\times3+2\times2$ .
Ask: Which parts must be doubled to count both rows?
Say: Each row contains both parts, so double both parts.
Used in: Repeated meal orders multiply every item’s cost, preventing an item from being counted only once.

If your child says…“I multiply only the first number inside.”

Ask: Do both rows contain the group of two?
What’s true: Every part belongs to each copy of the group.

A real problem

Calculating rods made multiplication easier.

Napier published his calculating-rod method in 1617.

The rods turned multiplication into a series of additions.

Users could multiply without recalling multiplication tables.

National Museums Scotland HBCSE–TIFR

Cardboard calculating strips face up beside a pencil and closed notebook.

Concept Lab · Common misconception

The tempting mistake in Arithmetic Expressions

Open standalone misconception page →

Must every operation be done left to right?

Wrong

8 + 2 × 3 = 30

Why it feels right. I started with the first operation I saw.

Right

Multiplication takes precedence over addition here.

Worked example

8 + 2 × 3

(8 + 2) × 3

The brackets change what gets repeated three times.

Two groups of 3 give 6.
Add 8 to get 14.
The expression (8 + 2) × 3 represents 30.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

What fills the blank in $13+4=\square+6$ ?

11. Both complete sides must equal 17.
Which is greater: $120\div3$ or $35+25$ ?

$35+25$ . Its value is 60, while the other gives 40.
How does $113-25$ compare with $112-24$ ?

They are equal. The extra starting amount matches the extra amount taken away.
Subtract the bracketed sum of fifteen and fifty-six from one hundred. What comes first?

15 plus 56. The amount subtracted includes both costs.
In two minus ten plus four times six, which complete term contains multiplication?

$4\times6$ . The product contributes one value to the sum.
When rearranging eighty-three minus fourteen as a sum, what must move with fourteen?

The minus sign. The term is negative, and moving it must preserve its value.
Start with two hundred minus the bracketed sum of forty and three. After removing brackets, is three added or subtracted?

Subtract it too. The whole sum is subtracted, including the 3.
Two people each buy items costing forty-three and twenty-four rupees. Which two products, left uncalculated, add to their total?

$2\times43+2\times24$ . Both costs occur twice.

15 more questions with answers

Teach Arithmetic Expressions in eight steps.

Why it matters