Small rules: repeating and growing

How can you explain the next shape?

A repetition returns to the same unit; growth changes by a stated amount. Say the rule first.

Place circle, square, circle, square. Grouping in twos reveals the repeated unit: circle, square. Shapes distinguish the objects without relying on color.

Now place groups of 2, 4 and 6 blocks. The count increases by two instead of returning to a previous group. Add two blocks to make the next group. Repetition and growth are different rules.

If the rule is “start at 2 and add two each time”, the counts are 2, 4, 6, 8. The step from 6 is written 6+2=8. Letter expressions and large numbers are not needed.

A few examples do not determine a unique rule. After 2 and 4, you could add two or double each number. Both rules fit the first two numbers, but the next values are 6 and 8.

In this lesson we state a rule and then check its examples. Explain whether the same action works on a new example. Describe what repeats rather than only guessing an answer.

A rule can also determine an output from an input. In the next story we ask what happens to the number put in, rather than what follows a previous term. We will distinguish order from correspondence.

Listen one paragraph at a time

1 / 6

2 / 6

3 / 6

4 / 6

5 / 6

6 / 6

Hold on to this distinction

The shapes repeat after two places; the growing counts do not return. State the rule even when examples look similar.

Change it and check

Change the start and increase, and compare four groups. Explain how this differs from repeating shapes.

Starting countIncrease each time

Repeating shapes: ○ □ · ○ □ · ○ □

2

4

6

8

Start at 2; add 2 each time: 2 → 4 → 6 → 8

We use a stated starting count and increment. A finite list of examples does not prove a unique rule.

Try with paper and objects

Lay circle-square cards twice. Separately make groups of 1, 3 and 5 blocks, then make the next group under the add-two rule.

Check and explain another example

Check your idea 1

The pair ○ □ repeats in ○ □ ○ □. Which shape comes next?

Check your idea 2

Start at 1 and add 3: 1, 4, 7. What comes next?

Apply it in a new setting

Do the two numbers 2, 4 force the next number to be 6?

Write an answer yourself

Return to the explanation

Compare with a solution

9 · number only

7 plus 2 gives 9.

Does your explanation include these ideas?

Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.

Point to one repeated unit.Name the increase.Check an example against the stated rule.

Continue, or pause here

Pair each input with an output using one stated rule.