Free Grade 2 Logic Worksheets

A teacher's guide to Grade 2 logic worksheets

By second grade, children can do something that marks a real change in thinking: they can hold two rules at once. In Grade 1, a puzzle asks, "What is the rule?" or "Apply this rule." In Grade 2, a puzzle may say, in effect, "There are two rules at work, one along the rows and one down the columns. Find the missing piece." That is a new level of reasoning, and it goes hand in hand with number work that is moving to bigger numbers, bigger steps and sometimes counting backward.

This page gathers the three Grade 2 puzzles: Complete the Pattern, a two-by-two grid of shapes; What Comes Next?, a growing number sequence with larger steps; and Pick the Next Number, a multiple-choice sequence that can run in either direction. This guide explains what each is asking, how to teach it, what to expect from children, and how these puzzles lead into the algebraic thinking of the years ahead. The Pick the Next Number puzzle has its own full guide on the Grade 2 number pattern page.

A Complete the Pattern page: a two-by-two grid with one empty cell, and three choices, A, B and C, for what belongs in it.

What Grade 2 adds

Compared with Grade 1, the Grade 2 puzzles ask for more in three ways.

• Two dimensions. The grid has rows and columns, and each carries a rule.
• Bigger numbers. Sequences reach into the thirties and forties, with steps that can be as large as twelve.
• More thinking in both directions. Sequences may go down as well as up.
• Choosing among tempting answers. In multiple-choice puzzles, the wrong answers are built to look plausible.

These are the first signs of what mathematicians call structure. A child is beginning to see that a pattern has parts, and that the parts follow rules.

Complete the Pattern

A two-by-two grid has three cells filled in and one blank. Below the grid are three choices, A, B and C. The child decides which one belongs in the blank.

The rules are simple and consistent. Each row follows one rule, and each column follows another.

• Along each row, the shape stays the same and the color changes.
• Down each column, the color stays the same and the shape changes.

Look at the sample. The top row shows a pink circle and a lilac circle. The shape is the same (circles), and the color changes (pink to lilac). The left column shows a pink circle above a pink square. The color is the same (pink), and the shape changes (circle to square). So the blank, which sits in the second row and the second column, must be a lilac square: a square, because it is in the square's row, and lilac, because it is in the lilac column.

The answer choices are built to test the reasoning. In the sample, A is a lilac square (correct), B is a pink square (right shape, wrong color) and C is a lilac circle (right color, wrong shape). Each wrong answer matches one of the two rules but not the other. A child who attends to only one feature will be tempted.

What a child learns.

• Two rules at once. The key lesson. Shape and color vary independently.
• Reading a grid. Looking along a row and down a column.
• Elimination. Ruling out choices by checking each rule.
• Visual attention. Noticing color and shape together.
• Early algebraic structure. A grid with a row rule and a column rule is, in miniature, a function of two variables.

How to introduce it.

1. Look at the top row. "What is the same? What is different?"
2. Say the row rule. "They are both circles. The color changes."
3. Look at the left column. "What is the same? What is different?"
4. Say the column rule. "They are both pink. The shape changes."
5. Find the blank. "It is in the square's row and the lilac column."
6. Choose. "So it is a lilac square."
7. Check. Test the choice against the row and the column.

Strategies.

• Cover and compare. Cover the grid except the top row, and say what changes.
• Name the two rules aloud. "Across: same shape. Down: same color."
• Eliminate. "Choice B is the wrong color. Choice C is the wrong shape."
• Draw it. Sketch the choice in the blank, and look at the whole grid.
• Check both ways. Look along the row, then down the column.

Common mistakes.

• Choosing the right shape but the wrong color, or vice versa. The child has used only one rule. Ask, "Check the row. Check the column."
• Choosing the one that "looks nice." Ask the child to say why.
• Copying a neighbor. The child puts the same thing as the cell next to it. Ask what changes across the row.
• Not seeing the rule at all. Go back to a simpler task: line up four blocks in a two-by-two, and ask what changes.

Extensions. Make your own grid with blocks or sticky notes. Make a three-by-three grid. Swap the rules, so that color changes across and shape changes down. Make a puzzle for a friend.

What Comes Next? at Grade 2

The Grade 2 growing sequence has the same format as in Grade 1: six rows of three numbers and a blank. But the numbers are larger, and the steps range more widely.

A Grade 2 What Comes Next page. The steps are larger, and the numbers go further up the number line.

Look at the rows on the sample.

In Grade 1, the steps run from 2 to 6. In Grade 2, they range up to 10, and the starting numbers can reach 20. That means more demanding mental arithmetic: adding 8 to 33 or 10 to 29 asks for strategies that Grade 1 does not need.

Strategies for the arithmetic.

• Adding ten. Change the tens digit. 29 plus 10 is 39.
• Adding nine or eight. Add ten, then take away one or two. 33 plus 8 is 33 plus 10 minus 2, or 41.
• Bridging through a ten. 17 plus 8: add 3 to reach 20, then 5 more, for 25.
• Using place value. 25 plus 8 is 20 plus 5 plus 8, or 20 plus 13.
• Using doubles. 4 plus 4 is 8. A step of 4 can be thought of as "add two twos."

Strategies for the rule.

• Subtract to find the step. 25 minus 17 is 8.
• Check the second gap. 33 minus 25 is also 8. Good.
• Say it aloud. "Seventeen, twenty-five, thirty-three. Up by eight."
• Count up in jumps. From 17 to 25: 3 to reach 20, 5 more.
• Use an open number line. Draw jumps between the numbers.

The step is the rule. The page may seem to be about arithmetic, but the real point is the rule, and the habit of finding it. A child who asks "how much does it change each time?" is thinking algebraically.

Pick the Next Number, in brief

The third puzzle is a multiple-choice sequence: three numbers, a blank, and three choices. In Grade 2, the numbers can be bigger, the steps larger, and, about a third of the time, the sequence runs downward, so the step is subtracted. The wrong choices are built to look plausible: one is a near miss, off by one or two, and the other applies the wrong step.

A full guide to the puzzle, including how to spot and avoid the traps, is on the Grade 2 number pattern page.

Sequences that count down

Not every sequence goes up. At Grade 2, a number pattern may run downward, and a child has to notice that the numbers are getting smaller and subtract the step. Take 40, 33, 26. Each number is 7 less than the one before: 40 minus 33 is 7, and 33 minus 26 is 7. So the next number is 26 minus 7, which is 19.

Counting back is almost always harder than counting on, and a few supports help.

• Notice the direction first. Before looking for the step, ask: "Are the numbers getting bigger or smaller?"
• Use a number line. Draw jumps to the left.
• Count back in chunks. To take 7 from 26, take 6 to reach 20, and then 1 more.
• Use a hundred chart. Go up a row for ten less.
• Check by going forward. 19 plus 7 is 26, so 19 is correct.
• Say it aloud. "Forty, thirty-three, twenty-six, nineteen."

A child who can run a sequence backward understands it much better than one who can only run it forward, because they have seen that the step is a relationship, not just a direction.

Mental arithmetic for bigger steps

The jump from Grade 1 to Grade 2 brings bigger numbers, and the arithmetic needs a few more tools. Here is a quick guide, with examples that match the kinds of rows on the pages.

These are the same ideas that appear in the Grade 2 math pages, where a child learns to work with tens and ones. A child who is comfortable with them will find that the logic puzzles feel light, and that the arithmetic does not get in the way of the thinking.

Two rules at once, beyond the page

The grid puzzle is a small model of a very common situation: two things varying independently. Children meet these everywhere, and pointing them out is a lovely way to extend the page.

• A menu. Choose a main (pasta or rice) and a sauce (red or white). How many combinations?
• A school uniform. Choose a shirt color and a pants color.
• A timetable. The day runs down, the subject runs across.
• A calendar. Weeks go down, days go across.
• A multiplication table. One number runs across, the other runs down.
• A map. Letters along the bottom and numbers up the side give a position.
• A board game. A grid with rows and columns, such as noughts and crosses.

A child who sees that a grid has a rule for the rows and a rule for the columns has found one of the most common structures in all of mathematics.

A sample twelve-minute session on sequences

A session with a seven-year-old, on a What Comes Next page.

1. Minute 0 to 2. Warm up. "Count by 4s from 4. Count by 3s from 5." Notice when the child falters, and stop there.
2. Minute 2 to 4. Look at the first row of the page: 17, 25, 33. Ask, "How much does it go up?" Let the child find the gap with jumps: 17 to 20 is 3, 20 to 25 is 5, so 8.
3. Minute 4 to 5. Check the second gap. 25 to 33 is 8. The rule holds. "So the next is 33 plus 8."
4. Minute 5 to 7. Add 8 to 33 together, with the "add ten, take away two" move. Write 41.
5. Minute 7 to 11. Let the child do three more rows alone.
6. Minute 11 to 12. Ask the child to choose a start and a step, and make a row for you.

A session like this puts the logic in the child's hands, with the arithmetic as support.

Making your own grids

Making a grid is much harder than solving one, and it is a wonderful sign of understanding. Here is how to make one with a child.

1. Choose two shapes, such as a circle and a square.
2. Choose two colors, such as red and blue.
3. Decide the rules. "Shape changes down. Color changes across."
4. Draw the grid. Four cells, with the shape and color in each.
5. Hide one cell. Cover it with a sticky note.
6. Add wrong choices. One with the right shape and wrong color, one with the wrong shape and right color.
7. Swap with a friend, and let them solve it.

A child who has built a puzzle knows exactly why each wrong choice is wrong. That is understanding.

For a challenge, try a three-by-three grid, or add a third feature such as size. A pattern with three rules, shape, color and size, is quite demanding, and many adults find it tricky.

Talking about reasoning

The most useful thing an adult can do is to ask for the reasoning. The best questions are short.

• "How did you decide?" Opens up the thinking.
• "What else could it be?" Looks at alternatives.
• "How could we check?" Introduces verification.
• "What if the rule were different?" Opens a new puzzle.
• "Can you make one like this?" Shifts to creating.
• "Why is that one wrong?" Explains the distractors.

A child who answers these in their own words is doing real mathematics, however small the puzzle.

From pattern to function

The three Grade 2 puzzles share a deeper idea: a rule that produces output from input. A sequence, a grid and a pick-the-next puzzle are all functions in disguise.

• In 15, 19, 23, 27, the rule is "add 4." The input is the position, and the output is the number.
• In the shape grid, the rule is "shape from the row, color from the column."
• In a descending sequence, the rule is "subtract a step."

You do not need to use the word function with a seven-year-old. But the experience of finding a rule and applying it is exactly what the word describes. A child who has done many of these will find algebra a familiar friend, not a stranger.

A routine for a puzzle session

A short, steady session works best.

1. Warm up. Count by 3s, 4s, 5s and 10s together, forward and backward.
2. Choose a puzzle. Start with the type the child is least confident with.
3. Do the first one together, thinking aloud.
4. Let the child do two or three alone.
5. Check. Ask for an explanation.
6. Look at a mistake. What was the child thinking?
7. Finish with a challenge. "I'll make a grid. You tell me what's missing."

Fifteen minutes is a good length for a seven-year-old.

A sample fifteen-minute session

A session with a seven-year-old on a Complete the Pattern page.

1. Minute 0 to 3. Warm up with real objects. Make a two-by-two with four blocks: a red circle, a blue circle, a red square, and an empty space. "What goes here?" Talk it through.
2. Minute 3 to 5. Look at the page. Ask the child to name the rule across the top row, and the rule down the left column.
3. Minute 5 to 8. Do the first grid together. Check each choice against both rules.
4. Minute 8 to 12. Let the child do the next two grids alone. When they answer, ask "How do you know?"
5. Minute 12 to 14. Check, and talk about any mistake.
6. Minute 14 to 15. The child makes up a grid for you, with a hidden answer.

This session mixes hands-on work, talk and independent solving, and it ends with the child as the puzzle-setter.

Common difficulties and what helps

A child uses only one rule in the grid. Ask them to check the row and then the column, one at a time.

A child cannot see the grid's structure. Use physical blocks, and rearrange them.

A child finds the step but adds wrongly. Practice the mental math tools, and revisit addition within 100.

A child counts the numbers, not the jumps. Use a number line and count the jumps.

A child is confused by a descending sequence. Say it aloud as "counting back," and use a number line going left.

A child guesses. Slow down, and ask them to explain.

A child is frustrated. Choose an easier puzzle, or work together on one. End on a success.

A child is bored. Move to the hardest type, or challenge them to make their own.

For teachers

A Grade 2 classroom can use these puzzles in many ways.

• Math warm-ups. A sequence on the board, and the class finds the next.
• Centers. A page, a pencil and a partner.
• Early finishers. A quiet, engaging task.
• Differentiation. Choose different numbered pages for different children.
• Discussion. Ask a child to explain a grid to the class.
• Assessment. A page shows whether a child can find a rule, and apply it.
• Link to math. Number sequences extend skip counting. See the Grade 2 number sense pages.

For parents

You do not need to be a mathematician.

• Read the instruction together.
• Ask, "What do you notice?"
• Let your child explain.
• Resist telling. The satisfaction belongs to them.
• Make up your own. Use blocks, coins or toys.
• Praise thinking. "You checked both rules."

Questions that come up

My child finds the grids easy but not the number sequences. That is common. The grids are visual, and the sequences need arithmetic. Practice the mental math tools.

Is a descending sequence harder? Usually yes. Many children find counting back more difficult than counting on. Use a number line.

My child is ready for more. Try Grade 3, where the puzzles involve decoding, classifying and deducing.

Do I need an answer key? The puzzles are designed to be checked from the page itself. Each has one right answer, and checking it against the rule shows whether it fits.

How often? Two or three times a week, for ten to fifteen minutes.

Should my child do all three puzzle types? Yes, for variety, but there is no rule. Follow interest.

A Grade 2 puzzle asks a child to hold two rules at once, to count in bigger steps, and to choose wisely among tempting answers. It is a lot, and it is the kind of work that makes a thinker.