5.1 Number Puzzles: Missing-Number Equations & the Equal Sign
Key Takeaways
CogAT Level 8 Number Puzzles has 14 untimed items (minimum attempted: 7) and a Riverside estimate of about 15 minutes.
Each Level 8 item is one equation with one missing number, and the child picks from four numbers; trains appear only at Levels 5/6 and 7.
Riverside's Level 8 practice puzzles run from 4 = 2 + ? to two-step items such as 8 = 6 + 5 − ? and 2 = 11 − 7 − ?.
Riverside's two named mistakes are using the wrong operation and choosing an answer before the calculation is finished.
Checking by putting the answer back (4 + 5 = 9, and the left side is 9) catches most arithmetic slips.
5.1 Number Puzzles: Missing-Number Equations & the Equal Sign
Quick Summary: At Level 8, Number Puzzles (Subtest 5) has 14 untimed items. Each item shows one equation with one missing number, such as 9 = 1 + 3 + ?, and four numbers to choose from. The child picks the number that makes both sides of the equal sign total the same amount. Riverside's Level 5/6 and Level 7 tests use a picture format with two trains; Level 8 moves to number sentences.
What Changes at Level 8
Number Puzzles is the second subtest of the Quantitative Battery, between Number Analogies and Number Series. Riverside's Directions for Online Administration describe two different formats across the primary levels:
| Level | Number Puzzles Format (Riverside) |
|---|---|
| 5/6 and 7 | Each question shows two trains. The child picks the answer picture that makes the second train carry the same number of objects as the first train. |
| 8 | Each question presents one equation in which one element is missing. The child selects the missing number. |
| 9 and up | Equations with one or more missing elements, sometimes with shapes standing in for numbers. |
Riverside's CogAT Product Guide makes the same point from the other direction: Level 8 includes Number Puzzles questions that follow the upper-level format. That matters for preparation. A second grader who has only practiced picture trains may freeze when the booklet shows digits, plus signs and minus signs. Number Puzzles is also the shortest subtest at Level 8, with 14 items (minimum attempted: 7) and a Riverside planning estimate of about 15 minutes.
What an Item Looks Like
Riverside's Level 8 Practice Activities show each puzzle as a row of boxes inside a shaded strip. A number sits to the left of the equal sign. The right side holds numbers joined by plus or minus signs, and one box contains a question mark. Four numbers appear underneath, each with an answer circle.
[ 9 ] = [ 1 ] + [ 3 ] + [ ? ]
4 5 6 7
o o o o
The direction a child hears is short. Online, the audio says to click the answer that makes the numbers on both sides of the equal sign total the same amount. The practice script asks, "Which number goes in the square with the question mark?"
Riverside's six Level 8 practice puzzles show the range of forms:
| Practice Puzzle | Missing Number | How the Riverside Script Solves It |
|---|---|---|
| 4 = 2 + ? | 2 | What do we add to 2 to make 4? |
| 5 = 1 + ? | 4 | What do we add to 1 to make 5? |
| 3 = 7 − ? | 4 | What do we subtract from 7 to make 3? |
| 9 = 1 + 3 + ? | 5 | 1 + 3 is 4; what do we add to 4 to make 9? |
| 8 = 6 + 5 − ? | 3 | 6 + 5 is 11; what do we subtract from 11 to make 8? |
| 2 = 11 − 7 − ? | 2 | 11 − 7 is 4; what do we subtract from 4 to make 2? |
Notice the progression: one addition, then one subtraction, then two steps on the right side. Every value is small, but the child must keep track of which operation comes last.
The Big Idea: The Equal Sign Means "Same Amount"
Most first- and second-grade worksheets show problems in one direction: 5 + 3 = __. Children who only see that layout often believe the equal sign means "write the answer next." A Level 8 puzzle reverses the layout. The total comes first and the unknown sits inside the right side, so a child who treats "=" as "the answer comes next" will add everything in sight.
Riverside's overview states the goal plainly: choose the answer that makes the numbers on both sides of the equal sign the same amount. Teach that sentence word for word. Then add a balance image if it helps: the left side and the right side must weigh the same, and the question mark is the missing weight.
A Four-Step Routine
Riverside recommends two strategies: calculate each side of the equal sign separately, and select the answer choice that makes both sides total the same amount. The routine below turns those two strategies into steps a seven-year-old can follow.
- Read the target. Say the number on the left side: "The left side is 9."
- Work the known part of the right side. Combine every number that is already there: "1 plus 3 is 4."
- Ask the missing-number question. If the last sign is plus: "What do I add to 4 to make 9?" If the last sign is minus: "What do I take away from 4 to make 2?"
- Check by putting the answer back. "4 plus 5 is 9, and the left side is 9. Both sides match."
Three ways to find the missing number
| Method | How It Works on 9 = 1 + 3 + ? | Best For |
|---|---|---|
| Count up | Start at 4 and count 5, 6, 7, 8, 9 on fingers: five fingers, so the answer is 5 | Missing addends |
| Think of the related fact | 9 − 4 = 5, because addition and subtraction undo each other | Children who know facts to 10 or 20 |
| Try each choice | Test 4, 5, 6 and 7 in the square; only 5 makes the right side equal 9 | Checking, or when stuck |
For subtraction, the same thinking runs backward. In 3 = 7 − ?, ask "7 take away what leaves 3?" Count back from 7 to 3 (6, 5, 4, 3): four steps, so the answer is 4. Then check: 7 − 4 = 3.
Trying each choice is a legitimate strategy because there are only four. It also catches arithmetic slips: if two choices seem to work, one of the calculations is wrong.
The Mistakes Riverside Names
Riverside's Level 8 teacher guide lists two common mistakes for this subtest. Two further traps follow from the item layout.
| Mistake | What It Looks Like on 9 = 1 + 3 + ? | Why It Happens | Fix |
|---|---|---|---|
| Answering before finishing the calculation (Riverside) | Adds 1 + 3, sees 4 among the choices, and picks it | The partial result is one of the choices | Always finish step 3 and check by substituting |
| Using the wrong operation (Riverside) | In 3 = 7 − ?, adds 3 + 7 and looks for 10 | Habit of adding whenever numbers appear | Point to the sign before the question mark and say it aloud in practice |
| Copying the target | Picks the choice that matches the left side | The left number feels like "the answer" | Remember the target is already on the page; the square needs the missing part |
| Adding every number on the page | 9 + 1 + 3 = 13 | Treating "=" as "the answer comes next" | Cover the left side, solve the right side, then compare |
Building the Skill at Home
- Use real objects first. Put 9 counters in one pile. Build "1 and 3" in another, then ask how many more the second pile needs. Write the matching number sentence under the piles.
- Move the unknown around. Practice 9 = 4 + ?, 9 = ? + 4 and ? = 4 + 5 so the child learns that the question mark can sit anywhere. The method (make both sides the same) never changes.
- Mix the signs. After plus-only puzzles, add puzzles that end in minus and puzzles with two steps, as in 8 = 6 + 5 − ?.
- Keep numbers friendly. Riverside's practice puzzles stay at 11 or below. Fact fluency to 20 is enough; the skill being measured is reasoning about equality, not speed with large numbers.
- Say the check sentence. "4 plus 5 is 9, and the left side is 9." Children who habitually check catch most of their own errors.
In the Level 8 practice puzzle 8 = 6 + 5 − ?, which number belongs in the question-mark square?
2
11
8
3
A child solves 9 = 2 + 3 + ? and chooses 5 because 2 + 3 = 5. Which mistake from Riverside's Level 8 teacher guide is this?
Choosing an answer before finishing the calculation
Reading the boxes from right to left instead of left to right
Using the wrong operation for the last step of the puzzle
Copying the number shown on the left of the equal sign
Which description matches Number Puzzles at Level 8?
Pairs of numbers; find the rule that changes each pair the same way
Two trains; pick the load that makes the second train match
One equation with one missing number to choose
A balance scale; pick the weight that keeps both pans level
Sections you finish are checked off in the contents.