5.2 Number Series: Bead Strings, Repeating Cycles & Growing Rules

5.2 Number Series: Bead Strings, Repeating Cycles & Growing Rules

Quick Summary: CogAT Level 8 Number Series (Subtest 6) has 18 untimed items. Each item shows several strings of beads on an abacus-style frame, and the child picks, from four strings, the one that comes next. Riverside's Level 8 practice items move from a repeating cycle, to a steady +1 pattern, to patterns that grow by 2 or that repeat each count twice. The single most useful habit is to turn every string into a number before looking for the rule.

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What a Number Series Item Looks Like

At Level 9 and above, Number Series shows a row of numerals such as 3, 6, 9, 12, ?. At Level 8 the same reasoning is shown with beads. Riverside's Level 8 Practice Activities describe each item this way:

• A box holds a row of vertical strings of beads, like the rods of an abacus. Riverside's examples show three to six strings in the box, with an empty place at the right where the next string belongs.
• A horizontal line runs across the box and across the answer strings. It separates the first five beads on a string from the rest, so a child can count "five and two more" instead of counting seven single beads.
• On some items, numbers are printed under the strings to show how many beads each one has.
• Below the box are four answer strings. The child fills in the circle under the string that continues the pattern.

Riverside notes that students who interpret the beads numerically rather than visually find it easier to solve the harder questions. That is the core coaching point for this subtest.

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The Pattern Families in Riverside's Practice Items

Riverside's six Level 8 practice questions are a compact map of what the subtest asks. Each one uses a different rule.

This section teaches the first three families. Section 5.3 covers the shrinking, interleaved and alternating-step patterns.

1. Repeating cycles

A repeating cycle is a short block of counts that starts over. In Riverside's first practice item the strings hold 4, 3, 2 beads and then 4, 3, 2 again, so the next string must hold 4. The practice script then contrasts this item with the next one: "Do these beads also repeat a pattern?" Asking that question every time is a good habit, because a repeating cycle and a growing series can start the same way.

• How to spot it: a count you have already seen comes back, in the same order.
• What to pick: the count that starts the block again (or the next count inside the block).

2. Steady growth (a constant step)

In a constant-step series, every string has the same number of beads more than the one before it: 3, 4, 5, 6 grows by 1; 2, 4, 6, 8 grows by 2. Find the step by comparing neighboring strings, check that it is the same everywhere, and add it once more.

• Check two or three steps, not one. The first step of 1, 2, 4, 5 is +1, but the second is +2. Children who check only the first step miss alternating rules.
• Odd starts are fine. 1, 3, 5, 7 grows by 2 just as 2, 4, 6, 8 does. Some children assume "counting by twos" must use even numbers.

3. Paired (doubled) steps

In Riverside's third practice item the strings hold 1, 1, 3, 3, 5, 5 beads. Each count appears twice before the series moves on, and the counts grow by 2. The script models the reasoning: the number of beads is getting larger, so the next string should have more than 5; it is increasing by 2, so the answer is 7, not 6. The 6-bead string is an attractive wrong answer because it is "one more."

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The "Count, Label, Compare" Routine

Because two answer strings may differ by only one bead, judging length by eye is unreliable. Teach this routine during practice:

1. Count each string and say its number. Use the five-bead line: "five and two is seven." During home practice, the child can write the count above each string.
2. Compare neighbors. Say each change aloud in practice: "4 to 3 is one less, 3 to 2 is one less, 2 to 4 jumps back up." A jump back to an earlier count signals a cycle.
3. Name the rule and predict the next count before looking at the answers. "It repeats 4, 3, 2, so the next one is 4."
4. Count every answer string. Pick the one with exactly the predicted number. If none matches, recount the box; if two seem to match, the rule is not precise enough.

On test day the child counts silently: talking is not part of a group administration, and children should follow the teacher's directions about marking the booklet. Practice the routine aloud at home, then silently in the last week before testing.

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The Mistakes Riverside Names

Riverside's Level 8 teacher guide lists three common mistakes for Number Series. Each one maps onto a specific wrong answer.

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Practice Ideas

• Bead and cube towers. Build strings with real beads or snap cubes, using one color for the first five, to mirror Riverside's five-bead line.
• Repeat or grow? Show the start of a series (3, 4, 5) and ask the child to make one version that repeats and one that keeps growing.
• Swap the representation. Give the same series as beads, as dots and as numerals. Children learn that the rule lives in the numbers, not in the picture.
• Short daily sets. Five or six mixed series a day, with the child explaining each rule, builds more than one long session.