A child sees and writes . Shown , the same child may say it is right. What does the equal sign really say?

Grade 1 · 1.OA.D.7

The equal sign means the same amount

A child sees 3 + 5 = \square and writes 8. Shown 3 + 5 = 8 + 2, the same child may say it is right. What does the equal sign really say?

Unlocks

Finding the missing number in an equationGrade 1

Where you are 23 in Counting and cardinality

Number9

Counting and cardinality23

▴ 3 above Saying the counting words to 20 Subitizing and ten frames

The equal sign means the same amount

Comparing quantities Counting a large pile by grouping ▾ Show all 23 (15 more)

Why this exists

Children meet the equal sign in sentences like 4 + 3 = 7, again and again, always with the sum on the left and one number on the right. From that steady diet a child can build a rule the teacher never meant: “the equal sign means now write the answer.” The rule works on every sentence they have seen so far, which is why it sticks.

It breaks the moment a sentence is written any other way. A child holding that rule reads 7 = 4 + 3 as backwards, reads 3 + 5 = 8 + 2 as “three and five is eight, then add two”, and later fills 8 + 4 = \square + 5 with 12. None of these is a hard sum. The trouble is only the sign.

This lesson gives the sign its real meaning early, while the numbers are small: the two sides name the same amount. That one idea is what lets a child find a missing number anywhere in a sentence, and years later it is the whole of algebra, which works by keeping two sides equal.

The lesson takes four days.

Part 1: the balance. The same amount in each pan keeps the beam level, and the equal sign says that in writing. One number on each side, or one number and a sum.
Part 2: true and false. An equation can be checked, and it can be false. The single number can sit on either side.
Part 3: a sum on each side, such as 6 + 3 = 5 + 4.
Part 4: deciding without finding both totals, by looking at how the two sides are built.

Definition

An equation is a number sentence with an equal sign, =, and something on each side of it. Each side names an amount: a single number such as 7, or a sum or difference such as 4 + 3 or 9 - 2.

The equal sign says: the amount on the left is the same as the amount on the right. Read it aloud as “is the same amount as”, or “equals”.

An equation is true when the two sides really do name the same amount, and false when they do not. So 7 = 4 + 3 is true, because 4 + 3 is 7. And 6 + 2 = 9 is false, because 6 + 2 is 8, which is not 9. A false equation is still an equation; it is simply a sentence that says something wrong, the way “a cat has six legs” is a sentence that is wrong.

The equal sign does not say which side to work out, and it does not say what to do next. It only says “same amount”.

Intuition

The picture that carries this lesson is a two-pan balance with linking cubes. Every cube weighs the same. Put cubes in both pans. If the two pans hold the same number of cubes, the beam sits level. If one pan holds more, that side goes down.

Figure 1The pans hold the same amount, so the beam is level. That is what 7 = 4 + 3 says.

Saara puts a train of 7 cubes in the left pan. In the right pan she puts a train made of 4 dark cubes and 3 light cubes. The beam is level. Written down, that is

7 = 4 + 3

The left pan is the left side of the equation and the right pan is the right side. The equal sign is the level beam. Nothing in the picture says “first” or “answer”: it shows two amounts that match.

Now Saara swaps the pans, so the 4 and 3 train is on the left. The beam is still level. That is 4 + 3 = 7, and it says the same thing. Either way round, the sign means “these two weigh the same”.

When the pans do not match, the beam tips, and there is no equal sign to write. A child who sees the beam go down on one side can say “that is false: those are not the same amount”. Saying which pan holds more, and writing a sign for it, waits for the comparison symbols; here the only question is “same or not the same?”

Dot cards give the same idea without the balance. Shilah lays dot cards on the table: a 6 card and a 3 card on one side of a pencil, a 5 card and a 4 card on the other. Counting each side gives 9 and 9. The pencil plays the part of the equal sign: the two groups of dots on either side of it are the same amount, even though they look different.

Results

Same amount means true. An equation is true when its two sides name the same number, and false when they do not. Working out each side separately settles it.

Either side can come first. If 4 + 3 = 7 is true, so is 7 = 4 + 3. Swapping the two sides never changes whether an equation is true, because “the same amount as” works both ways.

A number equals itself. 8 = 8 is true, and so is 4 + 9 = 4 + 9. A pan always balances its own copy.

A sum can sit on each side. 6 + 3 = 5 + 4 is true because both sides are 9. Two different ways of building the same amount are equal.

The sign is not a “next step”. In 3 + 5 = 8 + 2 the right side is all of 8 + 2, which is 10, not just the 8 next to the sign. Since 3 + 5 is 8, the equation is false.

Moving cubes from one part to the other keeps the amount. If one cube is taken from one part of a sum and joined to the other part, the total does not change. So 9 + 6 = 10 + 5 without adding either side all the way: one cube moved from the 6 to the 9.

Why it is true

Each side of an equation, worked out, is one number: the number of cubes in that pan. The equal sign is a claim that the two numbers are the same. So checking an equation is checking a claim, and that is why an equation can be true or false.

Why the order of the sides does not matter. “The left pan holds as many cubes as the right pan” and “the right pan holds as many cubes as the left pan” are the same fact told from the other side. Turning the balance around does not change what is in the pans.

Why a number equals itself. Two pans holding the very same count of cubes match cube for cube: pair each cube in one pan with a cube in the other and none is left over. That holds for any count at all, and for any sum, since 4 + 9 in one pan and 4 + 9 in the other are the same count.

Why two different sums can be equal. A sum names a number of cubes, not a particular way of snapping them together. A train of 6 dark and 3 light cubes and a train of 5 dark and 4 light cubes are both 9 cubes long, and the balance only feels how many cubes there are. So the equal sign compares amounts, not the way they were built.

Why moving a cube keeps the total. Take a cube off one part of a train and snap it onto the other part. No cube was added and none was lost, so the train has the same number of cubes. In numbers, 9 + 6 and 10 + 5 are the same cubes, regrouped, so they are the same amount.

Worked examples

Read each problem aloud and let the child build it with cubes or dot cards. Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.

Example 1

Seven against four and three

Saara puts 7 cubes in one pan. She builds a train of 4 cubes and 3 cubes and puts it in the other pan. Does the balance stay level? Write an equation with the 7 first. Is it true?

Solution

The right pan holds 4 cubes and 3 more. Counting on from 4: “five, six, seven.” That is 7 cubes, the same as the left pan, so the beam stays level (the picture in the Intuition section).

The 7 is in the left pan, so it goes on the left side:

7 = 4 + 3

This is true: the left side is 7 and the right side is 4 + 3 = 7. The single number is first and the sum is second, and that is fine. The equal sign only says the two pans match. Swapping the pans gives 4 + 3 = 7, the same true fact read from the other side.

Example 2

Eight against eight, and six against nine

Oskari puts 8 cubes in each pan. Then he empties the pans and puts 6 cubes in one and 9 in the other. For each, what does the balance do? Which of 8 = 8 and 6 = 9 is true?

Solution

Figure 2Eight against eight stays level. Six against nine tips toward the nine.

With 8 cubes in each pan, the beam stays level. Each cube in one pan can be paired with a cube in the other, with none left over. So

8 = 8

is true. It is a perfectly good equation even though there is nothing to add: a number is always the same amount as itself.

With 6 against 9, the beam tips down on the 9 side. The pans do not hold the same amount, so 6 = 9 is false. It is still an equation; it just says something that is not so.

Example 3

True or false, one number alone

Is each equation true or false?

(a) 12 = 7 + 5 (b) 6 + 2 = 9 (c) 8 = 13 - 5

Solution

Work out each side on its own, then ask “same amount?”

(a) The right side is 7 + 5. Making ten: 7 and 3 make 10, and 2 more is 12. The left side is 12. Both sides are 12, so 12 = 7 + 5 is true.

(b) The left side is 6 + 2 = 8. The right side is 9. Since 8 is not 9, the equation 6 + 2 = 9 is false. In a balance, 8 cubes against 9 would tip toward the 9.

(c) The right side is 13 - 5. Take 3 away to reach 10, then 2 more to reach 8. So 13 - 5 = 8, and the left side is 8. The equation 8 = 13 - 5 is true.

In (a) and (c) the number alone sits on the left. That changes nothing: the question is the same on every equation.

Example 4

Four ways to write one amount

Shilah has two dot cards, each with 4 dots, and one card with 8 dots. Which of these are true?

\begin{gathered} 4 + 4 = 8 \qquad 8 = 4 + 4 \\ 8 = 8 \qquad 4 + 4 = 4 + 4 \end{gathered}

Solution

All four are true. The two 4 cards together show 8 dots, the same as the 8 card, so 4 + 4 = 8 is true. The second equation is the same fact with its sides swapped, so 8 = 4 + 4 is true as well.

The third compares the 8 card with itself: 8 = 8 is true. The fourth puts a pair of 4 cards on each side. Each side is 8, so 4 + 4 = 4 + 4 is true too.

Only the first has the familiar shape, a sum and then its total. Read as “same amount”, the sign makes all four true, and that is the meaning to aim for.

Example 5

Six and three against five and four

Saara builds a train of 6 dark and 3 light cubes. Oskari builds a train of 5 dark and 4 light cubes. They put one train in each pan. Is 6 + 3 = 5 + 4 true?

Solution

Figure 3Different trains, the same amount. The beam stays level.

Work out each side by itself. Left: 6 + 3 = 9. Right: 5 + 4 = 9. Both sides are 9, so

6 + 3 = 5 + 4

is true, and the beam stays level.

The two trains look different: one has more dark cubes, the other more light ones. The equal sign does not care how a side is built, only how much it is. Laying the trains side by side shows it too: they are the same length.

Example 6

Three and five against eight and two

Is 3 + 5 = 8 + 2 true or false?

Solution

Figure 4The right pan holds more, so the beam tips. 3 + 5 = 8 + 2 is false.

The left side is 3 + 5 = 8. The right side is the whole of 8 + 2, which is 10. The two sides are 8 and 10, which are not the same amount, so the equation is false. In the balance, the pan with 8 + 2 goes down.

The 8 just after the sign matches the left side, but the right side is not that 8 alone; it is everything after the sign. Pointing at each side in turn and asking “how many cubes in this pan?” keeps the two sides apart. A true equation here would be 3 + 5 = 8 alone, or 3 + 5 = 6 + 2.

Example 7

Moving dots from one card to another

Shilah has dot cards showing 8 and 5 on one side, and 10 and 3 on the other. Is 8 + 5 = 10 + 3 true? Decide by moving dots, not by adding.

Solution

Figure 5Move two dots from the 5 card to fill the 8 card. No dot is added or lost.

The 8 card has room for 2 more dots to fill it to 10. Take 2 of the 5 dots and move them across. Now the first card shows 10 and the second shows 5 - 2 = 3. The left side has become 10 + 3, using the very same dots, with none added and none lost. So the two sides are the same amount, and

8 + 5 = 10 + 3

is true. Adding each side confirms it: 8 + 5 = 13 and 10 + 3 = 13.

This is the reason making ten works. Each time a child turns 8 + 5 into 10 + 3, they are using a true equation with a sum on each side.

Example 8

The same first part

Without adding, decide whether each is true or false.

(a) 7 + 6 = 7 + 5 (b) 4 + 9 = 4 + 9

Solution

(a) Picture two trains. Each starts with 7 dark cubes. The left train then has 6 light cubes and the right train 5. The dark parts match, so the whole question is whether 6 light cubes match 5, and they do not: the left train is one cube longer. The equation is false. Adding agrees: 7 + 6 = 13 and 7 + 5 = 12.

(b) Both sides are built from exactly the same parts in the same order. Two copies of one train are the same length, so 4 + 9 = 4 + 9 is true, whatever 4 + 9 turns out to be. (It is 13.)

Looking at how the sides are built, before counting, often settles an equation faster than adding. That habit is the start of reasoning with equations rather than calculating them.

Boundaries

The equal sign compares amounts, not looks. 6 + 3 and 5 + 4 look different and are equal. A pan of 4 red cubes and a pan of 4 blue cubes balance. Color, order and the way a train is snapped together do not matter; only how many.

A side is everything on that side. In 3 + 5 = 8 + 2 the right side is the whole of 8 + 2, not the 8 beside the sign. An equal sign joins two whole amounts.

More than one equal sign makes more than one claim. A sentence with two equal signs says that all three amounts are the same. 8 = 4 + 4 = 6 + 2 is true, since every part is 8. But 3 + 5 = 8 + 2 = 10 is false: its first sign claims 8 is the same amount as 10. Each equal sign must join two amounts that really match.

The balance needs equal cubes. A two-pan balance compares weight. It shows “same number” only because every linking cube weighs the same. With a mix of big and small objects, a level beam would not mean the same count.

Taking away needs enough to take from. With the numbers used here, a side such as 9 - 4 makes sense but 4 - 9 does not: there is no taking 9 cubes from 4. An equation with 4 - 9 on one side is not one to judge yet.

“False” is only “not the same”. This lesson asks one question: same amount, or not. Which side is greater, and the signs that say so, belong to the comparison symbols.

Where this goes next

The next step is finding a missing number in an equation. Once the sign means “same amount”, a box can sit anywhere, as in 8 + \square = 11 or \square = 6 + 7, and the child asks “what makes both sides the same?”

The same meaning sits under fact families, where 7 = 4 + 3 and 4 + 3 = 7 are one fact read from either side, and under the order and grouping of a sum, whose rules are themselves true equations with a sum on each side.

Much later the balance comes back almost unchanged. An equation, set against an expression, is a claim that two sides are equal, and solving one rests on the properties of equality: do the same thing to both pans and the beam stays level.

Exercises

Part 1: the balance

Saara puts 9 cubes in one pan and a train of 5 cubes and 4 cubes in the other. Does the balance stay level? Write an equation with the 9 first.
Oskari puts 7 cubes in each pan. Write the equation the balance shows. Is it true?
Is 5 = 8 true or false? What would a balance with 5 cubes and 8 cubes do?
Shilah puts a train of 6 cubes and 5 cubes in one pan and 12 cubes in the other. Does the balance stay level? Is 6 + 5 = 12 true?

Part 2: true or false

Is each equation true or false?

15 = 8 + 7
9 + 3 = 13
6 = 14 - 8
17 - 9 = 9
11 = 11
Write 5 + 9 = 14 with its two sides swapped. Is the new equation true?

Part 3: a sum on each side

Is each equation true or false?

7 + 2 = 6 + 3
4 + 6 = 5 + 4
9 + 7 = 10 + 6
2 + 6 = 8 + 1
12 - 4 = 3 + 5
Shilah lays out the dot cards below. Write her equation and say whether it is true.

Figure 6Dot cards for Exercise 16.

Part 4: without finding the totals

Is 6 + 8 = 6 + 8 true? Decide without adding.
Is 5 + 9 = 5 + 7 true? Decide without adding.
Is 7 + 8 = 10 + 5 true? Decide by moving cubes from one part to the other.
Is 9 + 6 = 8 + 7 true? Decide by moving one cube.
Write two different true equations that have 12 alone on one side and a sum on the other.

Explain

Read 4 + 5 = 9 + 3 aloud. Is it true? What does the equal sign say about the two sides?
Why is 10 = 10 true? Use a balance in your answer.
8 + 4 = 12 is true. Is 12 = 8 + 4 true? Explain why without adding.
Both 5 + 2 = 7 and 5 + 2 = 7 + 1 have 7 right after the equal sign. Why is one true and the other false?

Answers

True: yes, it stays level. 9 = 5 + 4, which is true: counting on from 5, “six, seven, eight, nine”, gives 5 + 4 = 9.
7 = 7. True: each cube in one pan pairs with a cube in the other.
False. 5 and 8 are not the same amount, so the balance tips down on the side with 8 cubes.
No, it tips toward the 12 cubes. False: 6 + 5 = 11 (a double, 5 + 5 = 10, and one more), and 11 is not 12, so 6 + 5 = 12 is false.
True. 8 + 7 = 15: 8 and 2 make 10, and 5 more is 15.
False. 9 + 3 = 12, and 12 is not 13.
True. 14 - 8 = 6: take 4 to reach 10, then 4 more to reach 6.
False. 17 - 9 = 8 (9 + 8 = 17), and 8 is not 9.
True. A number is the same amount as itself.
14 = 5 + 9. True: swapping the sides keeps the same amount on each side, and 5 + 9 = 14.
True. Both sides are 9: 7 + 2 = 9 and 6 + 3 = 9.
False. 4 + 6 = 10 but 5 + 4 = 9.
True. Both sides are 16: 9 + 7 = 16 and 10 + 6 = 16.
False. 2 + 6 = 8 but the right side is all of 8 + 1, which is 9.
True. Both sides are 8: 12 - 4 = 8 and 3 + 5 = 8.
7 + 4 = 5 + 6. True: both sides are 11, since 7 + 4 = 11 and 5 + 6 = 11.
True. Both sides are built from the same parts, so they are the same amount. (Each is 14.)
False. Both start with 5, but 9 is more than 7, so the left side is bigger. Adding agrees: 5 + 9 = 14 and 5 + 7 = 12.
True. Move 3 cubes from the 8 to the 7: the 7 becomes 10 and the 8 becomes 5, so 7 + 8 is the same cubes as 10 + 5. Both are 15.
True. Move one cube from the 9 to the 6: the 9 becomes 8 and the 6 becomes 7, so 9 + 6 is the same cubes as 8 + 7. Both are 15.
Many answers. Two of them: 12 = 6 + 6 and 12 = 10 + 2. Any two numbers that make 12 will do, with the 12 on either side.
False. Read aloud, it says “four plus five is the same amount as nine plus three.” It is false: the left side is 4 + 5 = 9 and the right side is 9 + 3 = 12. The equal sign says the two whole sides are the same amount; it does not say “the answer to 4 + 5 comes next”.
Put 10 cubes in each pan. Every cube in one pan pairs with a cube in the other and none is left over, so the beam stays level. Any number balances itself.
Yes. 8 + 4 = 12 says the left pan matches the right pan. 12 = 8 + 4 is the same balance turned around, and turning a balance around does not change what is in the pans.
In 5 + 2 = 7 the right side is 7, and 5 + 2 = 7, so the sides match. In 5 + 2 = 7 + 1 the right side is all of 7 + 1, which is 8, and 7 is not 8. What matters is the whole side, not the number next to the sign.