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Grade 2 Prep Guide

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Adaptive Math Test — Grade 2 Prep Guide

A comprehensive, topic-by-topic study guide for the Grade 2 adaptive math assessment. None of the practice problems below are taken from the test bank — they are fresh examples designed to teach the same skills.

Use this guide to:

  • Refresh the key idea behind each topic
  • Walk through fully-solved examples
  • Spot common mistakes before they cost you a question
  • Practice with answer keys at the end of every section

Table of Contents

Operations

  1. Addition within 100
  2. Subtraction within 100
  3. Addition within 1000
  4. Subtraction within 1000
  5. Mental Math

Place Value 6. Hundreds, Tens, and Ones 7. 3-Digit Numbers

Measurement 8. Ruler Use 9. Centimeters and Inches

Data and Graphs 10. Bar Graphs 11. Picture Graphs

Geometry 12. Partitioning Shapes 13. Faces, Edges, Vertices

Time and Money 14. Telling Time to 5 Minutes 15. Counting Coins and Dollars

Patterns and Logic 16. Skip Counting 17. Math Kangaroo Puzzles

1. Addition within 100

Key Concepts

Adding two-digit numbers uses place value: ones, tens. Strategies:

  • Add tens, then ones. 47 + 35 → 40 + 30 = 70; 7 + 5 = 12; 70 + 12 = 82.
  • Make a 10. 19 + 6 → 19 + 1 = 20; 20 + 5 = 25.
  • Round and adjust. 47 + 38 → 47 + 40 = 87; subtract 2 → 85.
  • Doubles and near-doubles. 26 + 27 → (26 + 26) + 1 = 53.

Worked Examples

Example 1. What is 36 + 21? Solution. Tens: 30 + 20 = 50. Ones: 6 + 1 = 7. Total: 57.

Example 2. What is 48 + 27? Solution. 48 + 20 = 68. 68 + 7 = 75.

Example 3. Sara has 42 stickers and gets 28 more. How many does she have? Solution. 42 + 28 = 70. (Tens: 40 + 20 = 60. Ones: 2 + 8 = 10. Total: 70.)

Example 4. What is 19 + 23? Solution. Make a 10: 19 + 1 = 20; 20 + 22 = 42.

Example 5. What is 39 + 39? Solution. Round and adjust: 40 + 40 = 80; subtract 2 → 78.

Common Mistakes

  • Forgetting to carry 1 to the tens when ones add to 10 or more.
  • Misaligning columns: writing 8 + 27 as 8 + 27 with the 8 under the 2 instead of the 7.
  • Adding tens to ones (treating 30 + 5 as 8 instead of 35).
  • Skipping the carry when one number is just slightly under a tens (e.g., 19 + 6).

Practice

  1. 24 + 39 = ?
  2. 17 + 56 = ?
  3. 65 + 28 = ?
  4. Tom has 19 marbles; Sam gives him 23 more. How many marbles does Tom have?
  5. 53 + 47 = ?

Answers: 1. 63 | 2. 73 | 3. 93 | 4. 42 | 5. 100

2. Subtraction within 100

Key Concepts

Subtraction is finding the difference. Two strategies:

  • Subtract by place value. 84 − 32 → 80 − 30 = 50; 4 − 2 = 2; 50 + 2 = 52.
  • Count up from the smaller number. 84 − 32 → from 32 to 80 is 48; 80 to 84 is 4; total 52.
  • Round and adjust. 65 − 28 → 65 − 30 = 35; add back 2 → 37.

When the ones digit of the bigger number is smaller, you regroup (borrow): take 10 from the tens.

Worked Examples

Example 1. What is 76 − 23? Solution. Tens: 70 − 20 = 50. Ones: 6 − 3 = 3. Total: 53.

Example 2. What is 52 − 18? Solution. Round: 52 − 20 = 32; add back 2 → 34. Or regroup: 4 tens 12 ones − 1 ten 8 ones = 3 tens 4 ones.

Example 3. What is 90 − 47? Solution. 90 − 50 = 40; add back 3 → 43.

Example 4. A bus had 60 passengers; 24 got off. How many remain? Solution. 60 − 24 = 36.

Example 5. What is 100 − 67? Solution. Count up: 67 → 70 is 3; 70 → 100 is 30; total 33.

Common Mistakes

  • "Subtracting up" within a column (writing 5 − 8 as 3 instead of regrouping).
  • Forgetting that after borrowing, the tens digit drops by 1.
  • Confusing direction in word problems (when "less", "fewer", or "remain" mean subtraction).
  • Not double-checking by adding back: if 84 − 39 = 45, then 45 + 39 should equal 84.

Practice

  1. 73 − 28 = ?
  2. 81 − 46 = ?
  3. 100 − 35 = ?
  4. A jar has 92 candies; 38 are eaten. How many remain?
  5. 60 − 27 = ?

Answers: 1. 45 | 2. 35 | 3. 65 | 4. 54 | 5. 33

3. Addition within 1000

Key Concepts

Add by place value: hundreds, tens, ones. When a column sums to 10 or more, carry to the next place. Mental tricks still help:

  • Round to friendly hundreds first.
  • Break apart one number (e.g., 287 + 156 → 287 + 100 + 50 + 6).

Worked Examples

Example 1. What is 234 + 152? Solution. Hundreds: 200 + 100 = 300. Tens: 30 + 50 = 80. Ones: 4 + 2 = 6. Total: 386.

Example 2. What is 376 + 248? Solution. 376 + 200 = 576. 576 + 48 = 624.

Example 3. A library has 459 fiction and 287 non-fiction books. How many books? Solution. 459 + 287 = 746. (459 + 300 = 759; subtract 13 → 746.)

Example 4. What is 198 + 365? Solution. Round: 200 + 365 = 565; subtract 2 → 563.

Example 5. A school sold 234 tickets on Monday, 156 on Tuesday, and 198 on Wednesday. How many in total? Solution. 234 + 156 = 390. 390 + 198 = 588.

Common Mistakes

  • Forgetting carries that cascade (e.g., 99 + 1 = 100 affects tens AND hundreds).
  • Lining up digits incorrectly in vertical layout.
  • Treating each column independently and missing the carry.

Practice

  1. 327 + 458 = ?
  2. 583 + 146 = ?
  3. 199 + 256 = ?
  4. A farmer harvests 478 apples on Saturday and 234 on Sunday. How many apples in total?
  5. 365 + 287 + 138 = ?

Answers: 1. 785 | 2. 729 | 3. 455 | 4. 712 | 5. 790

4. Subtraction within 1000

Key Concepts

Same place-value approach as 2-digit subtraction, but with hundreds. Three powerful strategies:

  • Place-value subtraction — subtract hundreds, tens, ones separately.
  • Round and adjust — 645 − 298 → 645 − 300 + 2 = 347.
  • Count up — from 298 to 645 = 2 + 45 + 300 = 347.

When a column needs regrouping, borrow from the next-larger place.

Worked Examples

Example 1. What is 478 − 156? Solution. Hundreds: 400 − 100 = 300. Tens: 70 − 50 = 20. Ones: 8 − 6 = 2. Total: 322.

Example 2. What is 524 − 287? Solution. 524 − 300 = 224. 224 + 13 = 237.

Example 3. What is 700 − 365? Solution. Count up: 365 → 400 is 35; 400 → 700 is 300; total 335.

Example 4. A farmer had 824 chickens; she sold 359. How many remain? Solution. 824 − 359 = 465. (824 − 400 = 424; 424 + 41 = 465.)

Example 5. What is 901 − 458? Solution. 901 − 500 = 401; 401 + 42 = 443.

Common Mistakes

  • Borrowing from a 0 (e.g., 500 − 137) without doing the cascade properly: 500 → 4 hundreds, 9 tens, 10 ones → 363.
  • Reversing operands in the head ("7 − 9" instead of "regroup, then 17 − 9").
  • Forgetting to update the place after a borrow.

Practice

  1. 654 − 289 = ?
  2. 800 − 347 = ?
  3. 532 − 178 = ?
  4. A bookstore had 945 books and sold 367. How many books are left?
  5. 1,000 − 458 = ?

Answers: 1. 365 | 2. 453 | 3. 354 | 4. 578 | 5. 542

5. Mental Math

Key Concepts

Mental math is faster math — done in your head with strategies, not by writing it out:

  • Friendly numbers. 28 + 19 → 28 + 20 − 1 = 47.
  • Compensation. 73 − 28 → 73 − 30 + 2 = 45.
  • Decomposition. 6 × 14 → 6 × 10 + 6 × 4 = 60 + 24 = 84.
  • Pairs that make 10. 7 + 9 + 3 → (7 + 3) + 9 = 19.
  • Double and halve (for multiplication). 4 × 25 → 2 × 50 = 100.

Worked Examples

Example 1. What is 38 + 27? Solution. Make a 10: 38 + 2 = 40; 40 + 25 = 65.

Example 2. What is 54 − 19? Solution. 54 − 20 = 34; add back 1 → 35.

Example 3. What is 12 + 17 + 8 + 3? Solution. Pair to make tens: (12 + 8) + (17 + 3) = 20 + 20 = 40.

Example 4. What is 5 × 18? Solution. Halve and double: 10 × 9 = 90. (Or: 5 × 20 − 5 × 2 = 100 − 10 = 90.)

Example 5. What is 100 − 47 − 13? Solution. Combine first: 47 + 13 = 60; 100 − 60 = 40.

Common Mistakes

  • Forgetting to "give back" what you added when rounding.
  • Trying to do too much at once in your head.
  • Reading numbers wrong while skipping a step (e.g., remembering 27 as 17).

Practice

  1. 46 + 39 = ?
  2. 80 − 27 = ?
  3. 25 + 25 + 25 + 25 = ?
  4. 7 + 8 + 13 + 12 = ?
  5. 6 × 15 = ?

Answers: 1. 85 | 2. 53 | 3. 100 | 4. 40 | 5. 90

6. Hundreds, Tens, and Ones

Key Concepts

A 3-digit number has three places:

  • Hundreds place — the leftmost digit times 100.
  • Tens place — the middle digit times 10.
  • Ones place — the rightmost digit times 1.

Example: in 528, the 5 means 500, the 2 means 20, the 8 means 8. Total: 500 + 20 + 8 = 528.

You can also have non-standard decompositions (e.g., 5 hundreds + 12 tens + 7 ones = 627), where one place "carries over" into the next.

Worked Examples

Example 1. What digit is in the tens place of 462? Solution. In 462, the digits go hundreds-tens-ones → 4-6-2. The tens digit is 6.

Example 2. What is the value of the 3 in 538? Solution. The 3 is in the tens place, so its value is 3 × 10 = 30.

Example 3. Build a number with 7 hundreds, 0 tens, 4 ones. Solution. 700 + 0 + 4 = 704.

Example 4. What number is 6 hundreds + 13 tens + 5 ones? Solution. 13 tens = 130. So 600 + 130 + 5 = 735.

Example 5. Write 219 in expanded form. Solution. 219 = 200 + 10 + 9 — that is 2 hundreds, 1 ten, and 9 ones.

Common Mistakes

  • Confusing the digit (e.g., "6") with its value (60 in the tens place).
  • Reading a number left to right but assigning places right to left, getting them flipped.
  • Forgetting that 12 tens needs to be regrouped as 1 hundred + 2 tens.

Practice

  1. What is the value of the 4 in 247?
  2. Write 583 as hundreds + tens + ones.
  3. What number equals 3 hundreds + 0 tens + 9 ones?
  4. What number equals 4 hundreds + 15 tens + 2 ones?
  5. In the number 670, what digit is in the tens place?

Answers: 1. 40 | 2. 500 + 80 + 3 | 3. 309 | 4. 552 | 5. 7

7. 3-Digit Numbers

Key Concepts

The smallest 3-digit number is 100 and the largest is 999. To compare 3-digit numbers, compare the hundreds first, then tens, then ones if needed.

You can also:

  • Add or subtract 10 or 100 quickly using place value (562 + 100 = 662; 562 − 10 = 552).
  • Round to the nearest ten or hundred. To round to the nearest 100, look at the tens digit: 5 or more → round up; less than 5 → round down.

Worked Examples

Example 1. Which is greater: 481 or 478? Solution. Hundreds equal (4). Tens equal (8). Ones: 1 < 8. So 478 < 481, meaning 481 is greater.

Example 2. What is 100 less than 845? Solution. Decrease the hundreds digit by 1: 845 − 100 = 745.

Example 3. Round 374 to the nearest 100. Solution. The tens digit is 7 (≥ 5), so round up to 400.

Example 4. Round 326 to the nearest 10. Solution. The ones digit is 6 (≥ 5), so round up to 330.

Example 5. What is the largest 3-digit number with digits 4, 8, and 2 (using each once)? Solution. Largest digit first: 842.

Common Mistakes

  • Comparing only the last digit (treating 481 < 478 because 1 < 8 — wrong).
  • Rounding 245 to 200 (looking at the ones place instead of the tens place).
  • Forgetting that "100 less" only changes the hundreds digit when no further regrouping is needed.

Practice

  1. Round 562 to the nearest 100.
  2. What is the smallest 3-digit number?
  3. Compare 749 and 794. Which is greater?
  4. 100 less than 612 = ?
  5. Round 588 to the nearest 10.

Answers: 1. 600 | 2. 100 | 3. 794 | 4. 512 | 5. 590

8. Ruler Use

Key Concepts

A ruler measures length. To measure with a ruler:

  1. Line up the 0 mark of the ruler with one end of the object.
  2. Read the number that the other end lines up with.
  3. Length = end position − start position.

If the object does not start at 0 (e.g., it starts at 3 cm and ends at 11 cm), the length is end − start = 11 − 3 = 8 cm.

Worked Examples

Example 1. A pencil starts at 0 cm and ends at 9 cm. How long is it? Solution. 9 − 0 = 9 cm.

Example 2. A crayon goes from the 2 cm mark to the 11 cm mark. How long is it? Solution. 11 − 2 = 9 cm.

Example 3. A piece of string is 14 cm. After 6 cm is cut off, how long is the rest? Solution. 14 − 6 = 8 cm.

Example 4. Two erasers placed end-to-end measure 7 cm + 5 cm. Total length? Solution. 7 + 5 = 12 cm.

Example 5. A ribbon is 18 cm long. How many 3-cm strips can be cut from it? Solution. 18 ÷ 3 = 6 strips.

Common Mistakes

  • Lining up the end of the ruler instead of the 0 mark (the very edge often has a small space before 0).
  • Reading from the wrong unit side (cm vs. inches).
  • Forgetting to subtract the start when an object doesn't begin at 0.

Practice

  1. A leaf goes from 1 cm to 8 cm on a ruler. How long is it?
  2. A pencil is 10 cm. After 4 cm is broken off, how much is left?
  3. Two sticks: one is 6 cm, one is 9 cm. End-to-end length?
  4. A book is 24 cm wide. A notebook is 17 cm wide. How much wider is the book?
  5. A ribbon is 30 cm. How many 6-cm pieces can be cut from it?

Answers: 1. 7 cm | 2. 6 cm | 3. 15 cm | 4. 7 cm | 5. 5 pieces

9. Centimeters and Inches

Key Concepts

Two common units for length:

  • Centimeter (cm) — used in metric system. Smaller unit.
  • Inch (in) — used in customary system. Larger unit.

Approximate equivalence: 1 inch ≈ 2.5 cm. So an inch is bigger than a centimeter.

For grade 2, you don't need exact conversion, but you should know:

  • Which unit is bigger (inch).
  • A pencil is roughly 7 inches or 18 cm.
  • A foot is 12 inches.

Worked Examples

Example 1. Which is longer: 1 inch or 1 centimeter? Solution. 1 inch is longer (about 2.5 cm).

Example 2. A book is 9 inches long. About how many centimeters is that (estimate)? Solution. 9 × 2.5 ≈ 22.5 cm.

Example 3. A nail is 5 cm. Is it more or less than 2 inches? Solution. 2 inches ≈ 5 cm, so the nail is about 2 inches (close enough).

Example 4. Which would you measure with inches: a desk or a fingernail? Solution. A desk uses inches (or feet); a fingernail uses centimeters or millimeters.

Example 5. Order from shortest to longest: 1 cm, 1 inch, 1 foot. Solution. 1 cm < 1 inch < 1 foot (since 1 foot = 12 inches).

Common Mistakes

  • Assuming the unit names are interchangeable (using "cm" and "inch" as if equal).
  • Estimating large object lengths in cm (most desks are easier to estimate in feet/inches).
  • Mixing up cm with mm.

Practice

  1. Which is longer: 5 cm or 5 inches?
  2. Estimate: a sheet of paper is about ___ inches long. (8, 11, or 20?)
  3. A pen is 14 cm. Is that more or less than 6 inches?
  4. Which unit fits a fingernail: inch or cm?
  5. Approximately how many cm in 4 inches?

Answers: 1. 5 inches | 2. 11 inches | 3. less than 6 inches (≈ 5.5 in) | 4. cm | 5. about 10 cm

10. Bar Graphs

Key Concepts

A bar graph uses bars of different heights/lengths to show how many of each category. To read one:

  1. Pick a category on the horizontal axis.
  2. Read the bar height on the vertical axis.
  3. Use the scale (each square / line = how many).

You can compare categories by looking at bar heights and answer "how many more", "how many in total", "which has the most", etc.

Worked Examples

Example 1. A bar graph shows: dogs = 6, cats = 4, birds = 3. How many pets in total? Solution. 6 + 4 + 3 = 13.

Example 2. Same graph as Example 1. How many more dogs than cats? Solution. 6 − 4 = 2 more dogs.

Example 3. A bar graph uses scale of 5. A bar reaches the 4 mark. Actual count? Solution. 4 × 5 = 20.

Example 4. Class A has 18 students; Class B has 23. Together? Solution. 18 + 23 = 41.

Example 5. Three bars: red 12, blue 8, green 16. Which has the fewest? Solution. Blue (8 is the smallest).

Common Mistakes

  • Reading the wrong axis (e.g., counting categories instead of values).
  • Ignoring the scale on a scaled graph (treating each square as 1 even when scale = 5).
  • Comparing visual bar height across graphs with different scales.

Practice

  1. A bar graph: apples = 9, oranges = 7, pears = 5. Total fruit?
  2. Same: how many more apples than pears?
  3. A bar graph (scale = 2): bar reaches 6 squares. Actual count?
  4. Three classes: 25, 30, 22. Which has the most?
  5. Two bars: 14 and 19. How many more than the smaller?

Answers: 1. 21 | 2. 4 | 3. 12 | 4. The class with 30 | 5. 5

11. Picture Graphs

Key Concepts

A picture graph uses symbols (★, 🍎, etc.) to show how many of each category. It always has a key that tells you what each symbol stands for (e.g., "each ★ = 1 book" or "each ★ = 5 books").

To find the count: count the symbols × value of one symbol.

Worked Examples

Example 1. Each ★ = 1 book. Lin's row has 6 stars. How many books did Lin read? Solution. 6 × 1 = 6 books.

Example 2. Each star = 4 books. A row has 3 stars. How many books? Solution. 3 × 4 = 12 books.

Example 3. Each ⚪ = 2 students. Class A has 5 circles. How many students in Class A? Solution. 5 × 2 = 10 students.

Example 4. Each flag = 3 votes. Pizza got 4 flags; burger got 2 flags. How many more votes for pizza? Solution. (4 − 2) × 3 = 2 × 3 = 6 more votes.

Example 5. Each ⛳ = 5 cars. Lot A has 3 symbols, Lot B has 5 symbols. Total cars? Solution. (3 + 5) × 5 = 8 × 5 = 40 cars.

Common Mistakes

  • Ignoring the key — counting each symbol as 1 when 1 symbol = 5.
  • Mis-counting partial symbols (sometimes a half-symbol means half the value).
  • Forgetting to multiply at the end.

Practice

  1. Each ★ = 2 stickers. A row has 7 stars. How many stickers?
  2. Each ⛳ = 3 students. A row has 4 flags. How many students?
  3. Each ⚪ = 1 pet. Three rows: 5, 3, 4 circles. Total pets?
  4. Each ★ = 5 votes. Apple: 3 stars. Banana: 6 stars. How many more banana votes?
  5. Each ★ = 4 cookies. Sara: 4 stars. Tom: 2 stars. How many cookies in total?

Answers: 1. 14 | 2. 12 | 3. 12 | 4. 15 | 5. 24

12. Partitioning Shapes

Key Concepts

To partition a shape means to cut it into equal pieces. Common partitions:

  • 2 equal parts — each part is a half (1/2).
  • 3 equal parts — each is a third (1/3).
  • 4 equal parts — each is a fourth or quarter (1/4).
  • 8 equal parts — each is an eighth (1/8).

The pieces must be equal in size to be called halves, thirds, etc.

Worked Examples

Example 1. A pizza is cut into 8 equal slices. What fraction is each slice? Solution. 1/8.

Example 2. A square is divided into 4 equal triangles. What fraction is each triangle? Solution. 1/4.

Example 3. A rectangle is split into 6 equal strips. If 2 are colored, what fraction is colored? Solution. 2/6 (which is the same value as 1/3).

Example 4. A circle is divided into 3 equal sectors. What fraction is one sector? Solution. 1/3.

Example 5. A square is split into 4 equal smaller squares. If 3 are filled in, what fraction is filled? Solution. 3/4.

Common Mistakes

  • Calling pieces "halves" when they aren't equal in size.
  • Confusing 3 parts (thirds) with 4 parts (fourths).
  • Counting the colored pieces but mistaking the total number of pieces.

Practice

  1. A bar is divided into 6 equal pieces. What fraction is one piece?
  2. A circle is in 4 equal parts; 1 is shaded. What fraction is shaded?
  3. A square is split into 8 equal triangles. Each triangle is what fraction?
  4. A rectangle is in 3 equal strips. 2 are colored. What fraction is colored?
  5. A pie is cut into 4 equal slices. How many slices is half the pie?

Answers: 1. 1/6 | 2. 1/4 | 3. 1/8 | 4. 2/3 | 5. 2 slices

13. Faces, Edges, Vertices

Key Concepts

Every 3-D shape has three things you can count:

  • Faces — the flat (or curved) surfaces.
  • Edges — the lines where two faces meet.
  • Vertices — the corners (where edges meet).
Shape Faces Edges Vertices
Cube 6 12 8
Rectangular prism 6 12 8
Triangular prism 5 9 6
Pyramid (square base) 5 8 5
Cylinder 3 (2 circles + 1 curved) 2 (curved) 0
Cone 2 (1 circle + 1 curved) 1 (curved) 1 (the tip)
Sphere 1 (curved) 0 0

Worked Examples

Example 1. Count the edges of a rectangular prism (a "box" shape). Solution. 4 edges around the top, 4 around the bottom, and 4 vertical edges connecting them = 12 edges.

Example 2. A tetrahedron (triangular pyramid) has how many vertices? Solution. 3 base corners + 1 apex on top = 4 vertices.

Example 3. A triangular prism has how many faces? Solution. 2 triangular ends + 3 rectangular sides = 5 faces.

Example 4. Does a sphere have any vertices? Solution. No — a sphere has 0 vertices (no corners).

Example 5. A cone has how many faces? Solution. 1 circular base + 1 curved side = 2 faces.

Common Mistakes

  • Confusing faces with edges (faces are surfaces; edges are lines).
  • Calling a curved surface "no face" — a cylinder still has a curved face.
  • Forgetting the apex of a pyramid as a vertex.

Practice

  1. How many vertices does a cube have?
  2. How many edges does a square pyramid have?
  3. A cylinder has how many vertices?
  4. How many edges does a triangular prism have?
  5. Which has more vertices: a square pyramid or a triangular prism?

Answers: 1. 8 | 2. 8 (4 base + 4 going up to the apex) | 3. 0 (curved surfaces only) | 4. 9 | 5. Triangular prism (6) > Square pyramid (5)

14. Telling Time to 5 Minutes

Key Concepts

A clock has two hands:

  • Hour hand (short) — points at the hour.
  • Minute hand (long) — points at how many minutes past the hour.

The clock face has 12 numbers, and each number represents 5 minutes (because 12 × 5 = 60). So when the minute hand points to the 3, it's 15 minutes past the hour.

Reading time:

  • Half past = 30 minutes (minute hand on 6).
  • Quarter past = 15 minutes.
  • Quarter to = 45 minutes (or "15 minutes before" the next hour).

Worked Examples

Example 1. The hour hand is on 4 and the minute hand is on 12. What time is it? Solution. 4:00.

Example 2. The hour hand is between 7 and 8; the minute hand is on 6. What time is it? Solution. 7:30.

Example 3. The minute hand is on 9. How many minutes past? Solution. 9 × 5 = 45 minutes.

Example 4. It's 5:20. Where do the hands point? Solution. Hour hand: a little past 5. Minute hand: on the 4 (since 4 × 5 = 20).

Example 5. What time is "quarter to 9"? Solution. 15 minutes before 9 = 8:45.

Common Mistakes

  • Mistaking the hour hand for the minute hand when they're close in length.
  • At 7:55, the hour hand is almost on 8 — the actual hour is still 7.
  • Reading 8:05 as "5:08" by mixing up the hands.

Practice

  1. The hour hand is on 6, minute hand on 12. What time?
  2. The minute hand is on 4. How many minutes past?
  3. It's 9:35. Which number is the minute hand on?
  4. Quarter past 5 = ?
  5. The hour hand is between 2 and 3, and the minute hand is on 9. What time?

Answers: 1. 6:00 | 2. 20 minutes | 3. 7 | 4. 5:15 | 5. 2:45

15. Counting Coins and Dollars

Key Concepts

Common US coin values:

  • Penny = 1¢
  • Nickel = 5¢
  • Dime = 10¢
  • Quarter = 25¢
  • Half-dollar = 50¢
  • Dollar = 100¢ = $1

To count a mix, add up by type and use the largest coins first to keep the running total simple.

Worked Examples

Example 1. 3 quarters and 2 dimes. How much? Solution. 3 × 25 + 2 × 10 = 75 + 20 = 95¢.

Example 2. 1 quarter, 4 dimes, 3 nickels. How much? Solution. 25 + 40 + 15 = 80¢.

Example 3. $1.00 + 2 quarters. Total? Solution. 100 + 50 = 150¢ = $1.50.

Example 4. Sara has 2 dimes and 6 pennies. How many cents? Solution. 20 + 6 = 26¢.

Example 5. Tom has 3 quarters and 1 dime. He spends 50¢. How much remains? Solution. He had 75 + 10 = 85¢; after spending 50: 35¢.

Common Mistakes

  • Mixing up dimes (10¢) and nickels (5¢) — dimes are smaller in size but bigger in value.
  • Forgetting that 1 dollar = 100 pennies = 4 quarters.
  • Counting coins twice.

Practice

  1. 2 quarters and 3 nickels = ?
  2. 4 dimes and 5 pennies = ?
  3. $1 + 2 quarters + 1 nickel = ?
  4. 6 nickels = ? cents
  5. You have 3 quarters and 2 dimes. You buy a snack for 65¢. How much remains?

Answers: 1. 65¢ | 2. 45¢ | 3. $1.55 (or 155¢) | 4. 30¢ | 5. 30¢

16. Skip Counting

Key Concepts

Skip counting is counting by a number greater than 1. It is the foundation of multiplication.

  • By 2s: 2, 4, 6, 8, 10, …
  • By 5s: 5, 10, 15, 20, 25, …
  • By 10s: 10, 20, 30, 40, …
  • By 100s: 100, 200, 300, …

Skip counting can start from any number: starting at 7 and counting by 5s → 7, 12, 17, 22, 27.

Worked Examples

Example 1. Count by 2s: 2, 4, 6, ___, ___. Solution. The pattern adds 2 each time → 8, 10.

Example 2. Count by 5s starting at 30: 30, 35, ___, ___. Solution. 40, 45.

Example 3. What is the 6th number in the count by 10s starting at 0? Solution. 0, 10, 20, 30, 40, 50 (the 6th term, counting 0 as the first).

Example 4. Count by 100s: 100, 200, 300, ___, ___, ___. Solution. 400, 500, 600.

Example 5. Count by 3s up to 24: how many numbers do you say? Solution. 3, 6, 9, 12, 15, 18, 21, 24 → 8 numbers.

Common Mistakes

  • Mixing the skip size (alternating between +2 and +3).
  • Starting at the wrong number (e.g., skipping by 5s but starting at 1 instead of 5).
  • Counting 0 as "the first term" inconsistently.

Practice

  1. Count by 2s: 14, 16, 18, ___, ___.
  2. Count by 5s starting at 20: 20, 25, 30, ___, ___.
  3. Count by 10s: 50, 60, 70, ___, ___.
  4. Count by 3s: 3, 6, 9, 12, ___, ___.
  5. Counting by 100s, what comes after 400?

Answers: 1. 20, 22 | 2. 35, 40 | 3. 80, 90 | 4. 15, 18 | 5. 500

17. Math Kangaroo Puzzles

Key Concepts

Math Kangaroo problems test logical thinking and clever arithmetic rather than memorized formulas. Useful tactics:

  • Try small cases first to spot a pattern.
  • Look for symmetry or balance.
  • Work backwards from the answer.
  • Eliminate impossible answers.
  • Count carefully — don't undercount the first or last item.

Worked Examples

Example 1. A bug climbs 3 steps up and 1 step down each minute. After 4 minutes, how many net steps up has it climbed? Solution. Each minute it gains 3 − 1 = 2 steps. After 4 minutes: 4 × 2 = 8 steps.

Example 2. Three friends each give one apple to the others. How many apples were given in total? Solution. Each friend gives 2 apples (one to each of the other two): 3 × 2 = 6 apples.

Example 3. A fence has 6 posts in a straight line. How many gaps are between the posts? Solution. For n posts in a line, there are n − 1 gaps: 6 − 1 = 5 gaps.

Example 4. I am thinking of a number. If you add 7 and then subtract 4, you get 12. What was my number? Solution. Work backwards: 12 + 4 − 7 = 9.

Example 5. A frog jumps forward 4 lily pads and back 2 lily pads each move. After 5 moves, how many lily pads forward is it? Solution. Net per move: 4 − 2 = 2 lily pads. After 5 moves: 5 × 2 = 10 lily pads.

Common Mistakes

  • "Off-by-one" errors in counting (gaps between posts, ages between siblings, etc.).
  • Reading the question too quickly and missing a detail.
  • Trying complicated math when a simple drawing or list would solve it.
  • Not checking the answer makes sense in context.

Practice

  1. There are 5 trees in a row. How many gaps are between trees?
  2. Four friends each shake hands with each other once. How many handshakes?
  3. I am thinking of a number. If you double it and add 5, you get 17. What's my number?
  4. A spider has 8 legs. How many legs do 6 spiders have together?
  5. A child counts 1, 2, 3, ... up to 30. How many times does the digit 2 appear?

Answers: 1. 4 | 2. 6 (4 × 3 ÷ 2) | 3. 6 | 4. 48 | 5. 13 (2, 12, 20, 21, 22 = 1+1+1+1+2 = uhh let me recount: 2, 12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29 — count "2"s in each: 2:1, 12:1, 20:1, 21:1, 22:2, 23:1, 24:1, 25:1, 26:1, 27:1, 28:1, 29:1 → 1+1+1+1+2+1+1+1+1+1+1+1 = 13)

Final Tips Before the Test

  • Read each question slowly — especially the operation word (more, fewer, total, share, each).
  • Underline key numbers in word problems.
  • Estimate first: if the problem looks like 47 + 28, you should expect an answer near 75. If you get 175, recheck.
  • Use scratch paper for any addition or subtraction with 2-digit or 3-digit numbers.
  • Check by working backwards when you have time.
  • Don't leave any blank — even a smart guess is better than nothing.
  • Manage your time — about 1 minute per question is a good pace.

Good luck — you've got this! 💪