Grade 5 (age 10-11)

grade 5 Patterns and Correspondence

Patterns and correspondence is about finding how one quantity changes as another changes, and writing that relationship as a formula. If one bottle holds 500 mL, then 2 bottles hold 1000 mL and 3 bottles hold 1500 mL: once you know the number of bottles, the total is fixed. In a table, look at how much △ grows each time ○ goes up by 1, and you can write the rule as a formula such as △ = ○ × 3 + 1. In the first picture you change the multiplier and the number added to see the table and the graph; in the second you change the amount in a bottle to see the total for each number of bottles.

How two quantities are related

🔗 When one is fixed, so is the other

①If one bottle holds 500 mL, 3 bottles hold 500 × 3 = 1500 mL

②Once you know the number of bottles, the total is fixed. A relationship like this, where one quantity changes as another changes, is called a correspondence

③Call the number of bottles ○ and the total △, and the relationship can be written as a single formula, △ = ○ × 500

Making a table

Multiply by3

Then add1

💡 Finding the rule in a table

①The top row of the table is ○ and the bottom row is the matching △. With the starting values, multiply by 3 and add 1, △ goes 4, 7, 10, 13 and so on

②Each time ○ goes up by 1, △ goes up by the multiplier, 3. That is why the points on the graph lie on a straight line

③Work back to ○ = 0 and you get 4 − 3 = 1, the number added; put together, the rule is △ = ○ × 3 + 1

④The 1 → 4, 2 → 7 and 3 → 10 under the table are what you get by putting ○ into this formula

Correspondence in everyday life

💡 Changing the amount per bottle

①The second picture is a correspondence with multiplication only and nothing added: total = number of bottles × amount per bottle

②With the starting 500 mL, 1 bottle is 500 mL, 2 bottles 1000 mL and 3 bottles 1500 mL, so each extra bottle adds 500 mL

③Change the amount per bottle and every bar stretches or shrinks in the same proportion, so the bar for ○ = 5 is always 5 times as long as the bar for ○ = 1

Amount per bottle500 mL

Summary

Correspondence

once ○ is fixed, △ is fixed

A relationship in which one quantity changes as another changes

The rule as a formula

△ = ○ × (multiplier) + (number added)

Seeing how much △ grows when ○ goes up by 1 gives you the multiplier

The rule on a graph

the matching points lie on a straight line

If △ always grows by the same amount, the line through the points is straight

🎯 At a glance

①Correspondence: once one quantity is fixed, the other is fixed too

②A table makes the rule easier to see

③Written as a formula, the rule gives values that are not in the table

④Bottles and total amount, items and price, time and distance: everyday life is full of correspondences

⑤Looking at the table and the graph together makes the rule clearer

Examples and Unit Test

Example 1

One bottle holds 500 mL. How many mL are there in 3 bottles?

Once the number of bottles (○) is fixed, the total (△) is fixed: this is a correspondence.

Total = bottles × 500, so 3 × 500 = 1500 (mL).

▸ 1500 mL

With the starting setting of the second picture, the bar for ○ = 3 shows 1500 mL.

Example 2

One bottle holds 500 mL. If ○ is the number of bottles and △ is the total, how do you write the correspondence as a formula?

Each time ○ goes up by 1, △ goes up by 500.

So the formula is △ = ○ × 500.

▸ △ = ○ × 500

When ○ is 0, △ is 0 too, so the formula needs only multiplication, with nothing added.

Unit Test

In △ = ○ × 500, what is △ when ○ = 5?

①505

②2500

③5500

④100

▸ ② 2500

Put ○ = 5 into △ = ○ × 500.

5 × 500 = 2500.

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Last updated 2026-10-17