Kid Einsteins · Multiplication Facts · Grades 3–4 · NYC Math Class
Multiplication Facts ×7, ×8, and ×9 — Area Models: 25 Practice Questions with Theory & Hidden Answers (Kid Einsteins)
A complete class-ready lesson on multiplication by 7, 8, and 9 taught with the area model: theory, patterns, doubling and 10-minus-1 shortcuts, and 25 practice questions with click-to-reveal step-by-step answers. Built for the SOMATH Kid Einsteins program (grades 3–4) on the Upper West Side of Manhattan.
This is Class 3 of the SOMATH Kid Einsteins multiplication arc — the class where the last three tricky times tables click: ×7, ×8, and ×9. We teach them using area models, because the picture of a rectangle is the same picture the student will see again in fractions, in fifth-grade area, in algebra tiles, and eventually in polynomial multiplication. Learn it once, use it forever.
Every question below has a hidden button that reveals the answer and the reasoning, so a student can practice honestly and then check their thinking. Written by the same team that teaches Kid Einsteins at SOMATH, a math-focused school on the Upper West Side of NYC run by cofounder Marcelo Ambrozio (Northwestern) and cofounder Vivianne Wright (Harvard).
- Warm-up: read the area-model theory aloud and draw one rectangle together (10 min).
- Skip-counting chant for 7, 8, and 9 (5 min).
- Students attempt the 25 questions with the answers hidden (20 min).
- Reveal answers as a group and re-teach any that missed 3+ students (10 min).
What’s in this lesson
- What an area model actually is
- Area models and the distributive property
- Skip counting — the shortcut to the times tables
- Theory: multiplying by 7 (double + triple)
- Theory: multiplying by 8 (double three times)
- Theory: multiplying by 9 (10 minus 1)
- Splitting the rectangle — the universal strategy
- 25 practice questions
- Answer key summary
- About SOMATH & Kid Einsteins
1. What an area model actually is
A multiplication fact is a rectangle
An area model is a rectangle drawn on grid paper (or imagined). One side of the rectangle is the first number in the multiplication; the other side is the second number. The number of unit squares inside is the answer.
Example: 6 × 7 is a rectangle that is 6 wide and 7 tall. It contains 42 unit squares, so 6 × 7 = 42.
| 6 | |||||
| ■ | ■ | ■ | ■ | ■ | ■ |
| ■ | ■ | ■ | ■ | ■ | ■ |
| ■ | ■ | ■ | ■ | ■ | ■ |
| ■ | ■ | ■ | ■ | ■ | ■ |
| ■ | ■ | ■ | ■ | ■ | ■ |
| ■ | ■ | ■ | ■ | ■ | ■ |
| ■ | ■ | ■ | ■ | ■ | ■ |
A 6-by-7 rectangle has 42 unit squares. That’s 6 × 7 = 42.
Every multiplication fact has an area-model picture. If a student can draw the rectangle, they understand what the multiplication means — not just what the answer is.
2. Area models and the distributive property
Split the rectangle — add the pieces
The whole trick of the area model is: if a rectangle is hard, chop it into two easier rectangles. This is the distributive property, drawn as a picture.
Example: 7 × 8 is tricky. Split the 8 side into 5 + 3:
| 5 | 3 | ||||||
| ■ | ■ | ■ | ■ | ■ | ■ | ■ | ■ |
| ■ | ■ | ■ | ■ | ■ | ■ | ■ | ■ |
| ■ | ■ | ■ | ■ | ■ | ■ | ■ | ■ |
| ■ | ■ | ■ | ■ | ■ | ■ | ■ | ■ |
| ■ | ■ | ■ | ■ | ■ | ■ | ■ | ■ |
| ■ | ■ | ■ | ■ | ■ | ■ | ■ | ■ |
| ■ | ■ | ■ | ■ | ■ | ■ | ■ | ■ |
Green piece: 7 × 5 = 35. Gold piece: 7 × 3 = 21. Total: 35 + 21 = 56. So 7 × 8 = 56.
Written as an equation, this is the distributive property:
7 × 8 = 7 × (5 + 3) = (7 × 5) + (7 × 3) = 35 + 21 = 56Every tough fact in this class can be handled with a split like this.
3. Skip counting — the shortcut to the times tables
Counting by 7s, 8s, and 9s
Skip counting means jumping forward by the same number each time. Every jump is one group; every number you land on is a multiple.
Counting by 7:
Counting by 8:
Counting by 9:
Notice 56 shows up in both the ×7 and ×8 lists (that’s 7 × 8), and 63 shows up in both the ×7 and ×9 lists (that’s 7 × 9). Common multiples are not accidents — they are the products you already know.
4. Theory: multiplying by 7 (double + triple)
Times 7 = the hardest table, made easy by splitting
7 is the trickiest single-digit multiplier because it doesn’t double cleanly. The best trick is to split 7 into 5 + 2 and use the area model:
Example: 7 × 8 = (5 × 8) + (2 × 8) = 40 + 16 = 56
The full ×7 table:
| Fact | Product | Fact | Product |
|---|---|---|---|
| 7 × 1 | 7 | 7 × 6 | 42 |
| 7 × 2 | 14 | 7 × 7 | 49 |
| 7 × 3 | 21 | 7 × 8 | 56 |
| 7 × 4 | 28 | 7 × 9 | 63 |
| 7 × 5 | 35 | 7 × 10 | 70 |
Patterns to notice:
- The ones digits of the ×7 table cycle: 7, 4, 1, 8, 5, 2, 9, 6, 3, 0. Every digit shows up exactly once in the first ten multiples.
- 7 × 7 = 49 is the “square” and worth memorizing on its own.
5. Theory: multiplying by 8 (double three times)
Times 8 = double, double, double
Multiplying by 8 is the same as doubling three times in a row, because 8 = 2 × 2 × 2.
Example: 7 × 8 = 7 → 14 → 28 → 56
That’s three doubles: 7 × 2 = 14, 14 × 2 = 28, 28 × 2 = 56. If a student knows their doubles, they know the ×8 table.
The full ×8 table:
| Fact | Product | Fact | Product |
|---|---|---|---|
| 8 × 1 | 8 | 8 × 6 | 48 |
| 8 × 2 | 16 | 8 × 7 | 56 |
| 8 × 3 | 24 | 8 × 8 | 64 |
| 8 × 4 | 32 | 8 × 9 | 72 |
| 8 × 5 | 40 | 8 × 10 | 80 |
Pattern to notice: Every multiple of 8 is even, and the ones digits cycle: 8, 6, 4, 2, 0, 8, 6, 4, 2, 0… The same five endings repeat forever.
6. Theory: multiplying by 9 (10 minus 1)
Times 9 = times 10, minus one copy
Multiplying by 9 is the easiest of the three once you know the trick: multiply by 10, then subtract one copy of the other number. That’s because 9 = 10 − 1.
Example: 7 × 9 = (7 × 10) − 7 = 70 − 7 = 63
As an area model, this is a 7-by-10 rectangle with one 7-tall column removed.
The full ×9 table:
| Fact | Product | Fact | Product |
|---|---|---|---|
| 9 × 1 | 9 | 9 × 6 | 54 |
| 9 × 2 | 18 | 9 × 7 | 63 |
| 9 × 3 | 27 | 9 × 8 | 72 |
| 9 × 4 | 36 | 9 × 9 | 81 |
| 9 × 5 | 45 | 9 × 10 | 90 |
Two beautiful patterns:
- The digits of every product add to 9. 27 → 2 + 7 = 9. 54 → 5 + 4 = 9. 72 → 7 + 2 = 9. Always.
- The tens digit is one less than the number being multiplied. 9 × 4 = 36 (tens digit 3 = 4 − 1). 9 × 7 = 63 (tens digit 6 = 7 − 1). That’s the finger-counting trick your parents used.
7. Splitting the rectangle — the universal strategy
Any tough fact = two easier facts
Here’s the single most important idea in this lesson, said out loud one more time:
Any multiplication fact can be split into two easier facts using the area model.
Example 1: 8 × 6. Split 6 into 5 + 1:
8 × 6 = (8 × 5) + (8 × 1) = 40 + 8 = 48Example 2: 9 × 8. Use the 10 minus 1 split:
9 × 8 = (10 × 8) − (1 × 8) = 80 − 8 = 72Example 3: 7 × 9. Split 9 into 10 − 1 from the ×9 side:
7 × 9 = (7 × 10) − (7 × 1) = 70 − 7 = 63This is the same picture — a rectangle, cut into two easier rectangles — that a student will use in fifth grade when they multiply 23 × 47 with the box method, in seventh grade when they distribute 3(x + 4), and in tenth grade when they FOIL (x + 2)(x + 5). Learn to see it now, and every future multiplication is a familiar picture.
25 Practice Word Problems
All 25 questions are word problems that use ×7, ×8, and ×9 with area-model thinking. Each one has a hidden button that reveals the answer and step-by-step reasoning.
Question 1
Ms. Chen’s classroom has 7 tables and 6 chairs at each table. She sets out one worksheet per chair. How many worksheets does she need?
Answer: 42 worksheets
Draw the area model: 7 rows (tables) × 6 columns (chairs) = 7 × 6 = 42. Split 7 into 5 + 2: (5 × 6) + (2 × 6) = 30 + 12 = 42.
Question 2
A pack of colored pencils has 8 pencils. Diego buys 4 packs for art class. How many colored pencils does he have?
Answer: 32 pencils
4 packs × 8 pencils each = 8 × 4 = 32. Double three times: 4 → 8 → 16 → 32.
Question 3
A bookshelf has 9 shelves, and each shelf holds 6 books. How many books fit on the whole bookshelf?
Answer: 54 books
9 shelves × 6 books each = 9 × 6 = 54. Use 10 minus 1: (10 × 6) − (1 × 6) = 60 − 6 = 54. Digit check: 5 + 4 = 9 ✓.
Question 4
A parking lot has 7 rows of parking spots with 8 spots in each row. How many cars can park there in total?
Answer: 56 cars
Area model: 7 rows × 8 columns = 7 × 8 = 56. Split: (5 × 8) + (2 × 8) = 40 + 16 = 56. Or double: 7 → 14 → 28 → 56.
Question 5
A tile floor is arranged in a square: 9 tiles across and 9 tiles down. How many tiles cover the floor?
Answer: 81 tiles
Area of a square = side × side = 9 × 9 = 81. Use 10 minus 1: (10 × 9) − (1 × 9) = 90 − 9 = 81. Digit check: 8 + 1 = 9 ✓.
Question 6
A gardener plants 49 tomato plants in a square patch with the same number of plants in every row. How many rows does she plant?
Answer: 7 rows
A square patch means rows = plants per row. Ask: what number times itself equals 49? 7 × 7 = 49. So she plants 7 rows of 7 plants each. (49 is the ×7 square.)
Question 7
The SOMATH front desk has 72 lollipops to hand out. They put them into bags of 8. How many bags do they fill?
Answer: 9 bags
Ask: 8 times what equals 72? From the ×8 table: 8 × 9 = 72. So they fill 9 bags. Check by doubling: 9 → 18 → 36 → 72 ✓.
Question 8
A soccer coach lines up 63 players in equal rows of 9. How many rows are there?
Answer: 7 rows
Ask: 9 times what equals 63? Use the tens-digit trick for ×9: the tens digit of 63 is 6, so the missing factor is one more — 7. Check: 9 × 7 = 63 ✓.
Question 9
A library has bookcases arranged in 7 rows with 8 bookcases per row. The janitor cleans 3 bookcases before lunch. How many are left to clean?
Answer: 53 bookcases
Total: 7 × 8 = 56 bookcases. Subtract the 3 already cleaned: 56 − 3 = 53. Two steps: multiply first, then subtract.
Question 10
A grocery store gets 9 cartons of orange juice, and each carton holds 8 bottles. If 5 bottles break during unloading, how many are still good?
Answer: 67 bottles
Total bottles: 9 × 8 = 72. Subtract the broken ones: 72 − 5 = 67 bottles still good. (Use 10 minus 1 for the ×9: 80 − 8 = 72.)
Question 11
A quilt is made of 7 rows of 9 squares. In each row there are 5 blue squares and 4 yellow squares. How many squares are in the whole quilt?
Answer: 63 squares
Split each row into blue + yellow, then use the area model:
7 × 9 = (7 × 5) + (7 × 4) = 35 + 28 = 6335 blue squares and 28 yellow squares. Both pieces are on the ×7 table you already know.
Question 12
A recycling bin holds 9 bundles of newspaper, and each bundle has 8 newspapers. Use the “10 minus 1” trick to find how many newspapers are in the bin.
Answer: 72 newspapers
Rewrite 9 as 10 − 1:
9 × 8 = (10 × 8) − (1 × 8) = 80 − 8 = 72Think of it as 10 bundles minus 1 bundle: 80 newspapers minus 8 newspapers = 72.
Question 13
Aisha bakes 7 trays with 8 cookies each. Ben bakes 9 trays with 6 cookies each. Who baked more cookies, and by how many?
Answer: Aisha baked more, by 2 cookies
Aisha: 7 × 8 = 56. Ben: 9 × 6 = 54. Aisha has more: 56 − 54 = 2 more cookies. Even though Ben used more trays, Aisha’s trays were bigger.
Question 14
A crate holds 8 rows of oranges with 6 oranges in each row. Solve using doubling three times.
Answer: 48 oranges
Area model: 8 × 6 = 48. Take 6 and double it three times: 6 → 12 → 24 → 48. So the crate holds 48 oranges.
Question 15
Mr. Ramos builds a square vegetable bed that is 7 feet on every side. He wants to buy enough soil to cover the area. How many square feet of soil does he need?
Answer: 49 square feet
Area of a square = side × side = 7 × 7 = 49 square feet. This is why 7 × 7 is called a “square” number — it’s literally the area of a square with side 7.
Question 16
A teacher is packing equal groups of 9 crayons into pencil cases. She has these counts written on the board: 45, 63, 74, 81. Only one of these amounts cannot be packed into full groups of 9 with none left over. Which number is it?
Answer: 74
Use the digit-sum trick for 9. 45 → 4+5 = 9 ✓ (that’s 9 × 5). 63 → 6+3 = 9 ✓ (9 × 7). 74 → 7+4 = 11 ✗. 81 → 8+1 = 9 ✓ (9 × 9). Only 74 is not a multiple of 9, so it can’t be split into full groups of 9.
Question 17
A movie theater has 9 rows of 7 seats. On Saturday night every seat is filled except for 4 empty ones in the back. How many people are watching the movie?
Answer: 59 people
Total seats: 9 × 7 = 63 (use 10 minus 1: 70 − 7 = 63). Subtract the 4 empty seats: 63 − 4 = 59 people.
Question 18
Camila is choosing a snack pack. Pack A has 8 bags with 9 pretzels each. Pack B has 7 bags with 10 pretzels each. Which pack has more pretzels, and by how many?
Answer: Pack A has more, by 2 pretzels
Pack A: 8 × 9 = 72. Pack B: 7 × 10 = 70. So Pack A has 2 more pretzels. Even though 10 looks like a bigger factor, the other factor (7) is small — the product depends on both.
Question 19
SOMATH orders 7 pizzas for the Kid Einsteins party, and each pizza is cut into 8 slices. If 48 slices are eaten, how many slices are left over?
Answer: 8 slices
Total slices: 7 × 8 = 56. Subtract the eaten slices: 56 − 48 = 8 slices left. Area-model split for 7 × 8: (5 × 8) + (2 × 8) = 40 + 16 = 56.
Question 20
Lucas has 45 stickers and wants to put them into an album that shows 9 stickers on each page. How many pages will he fill?
Answer: 5 pages
Ask: 9 times what equals 45? From the ×9 table: 5 × 9 = 45. So he fills 5 pages. Digit check on 45: 4 + 5 = 9 ✓.
Question 21
Amara has 7 pages in her photo album, and each page holds 8 pictures. She wants to add 10 more pictures at the end. How many pictures will the album have in total?
Answer: 66 pictures
Filled pages: 7 × 8 = 56. Add the new pictures: 56 + 10 = 66 pictures in total.
Question 22
A spider has 8 legs. A tank at the zoo has 9 spiders and 3 beetles (each beetle has 6 legs). How many legs are there in the tank in total?
Answer: 90 legs
Spider legs: 9 × 8 = 72 (10 minus 1: 80 − 8 = 72). Beetle legs: 3 × 6 = 18. Total: 72 + 18 = 90 legs.
Question 23
A rectangular rug is 9 feet long and 7 feet wide. A carpet cleaner charges one dollar per square foot. How much will it cost to clean the whole rug?
Answer: 63 dollars
Area = length × width = 9 × 7 = 63 square feet. At $1 per square foot, the cost is $63. The area model isn’t just a memorization trick — it’s literally how we compute area in real life.
Question 24
A school theater has 7 rows of seats with 12 seats in each row. Use the area model (splitting 12 into 10 + 2) to find the total number of seats.
Answer: 84 seats
Split the 12 side into 10 + 2:
7 × 12 = (7 × 10) + (7 × 2) = 70 + 14 = 84This is exactly how we’ll multiply 23 × 47 later in the year: split the big number, multiply the easier pieces, add them up.
Question 25
Challenge: Mr. Ramos has a rectangular garden that is 8 feet by 9 feet. He extends one of the 8-foot sides by adding a 1-foot-wide path. What is the total area of the new bigger rectangle?
Answer: 80 square feet
The original garden is 8 × 9 = 72 square feet.
The path adds a rectangle of 8 × 1 = 8 square feet.
New total area = 72 + 8 = 80 square feet.
You can also see this as one bigger rectangle: 8 × (9 + 1) = 8 × 10 = 80. That’s the distributive property again — adding onto a rectangle is the same as multiplying by a bigger side length.
Answer key summary
| Q# | Answer | Q# | Answer | Q# | Answer |
|---|---|---|---|---|---|
| 1 | 42 | 10 | 67 | 19 | 8 |
| 2 | 32 | 11 | 63 | 20 | 5 |
| 3 | 54 | 12 | 72 | 21 | 66 |
| 4 | 56 | 13 | Aisha, by 2 | 22 | 90 |
| 5 | 81 | 14 | 48 | 23 | $63 |
| 6 | 7 | 15 | 49 | 24 | 84 |
| 7 | 9 | 16 | 74 | 25 | 80 |
| 8 | 7 | 17 | 59 | ||
| 9 | 53 | 18 | Pack A, by 2 | ||
| 9 | 56 | 18 | 8 × 9 |
About SOMATH & Kid Einsteins
SOMATH — School of Math is a math-focused school on the Upper West Side of Manhattan for students in grades 1–12. The Kid Einsteins program (grades 3–4) drills multiplication and division fluency, fractions and decimals, area and perimeter, ratios, and structured word problems in small groups of 6–8 students. This is Class 3 of the Kid Einsteins arc.
Classes are taught by cofounder Marcelo Ambrozio (Northwestern-trained, 15+ years teaching math in NYC) and the SOMATH team.
Location: 226 W 79th St, 1st Floor, New York, NY 10024 (Upper West Side)
Phone: (646) 668-6151
Email: hello@schoolofmath.us
Book a free evaluation for your child — we’ll assess where they are and place them in the right small group.