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Count groups several equivalent ways · 10 practice problems

2.OA.C.43.OA.A.1

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Variant 1 easy

Find the choice that is not a correct way to count how many beads there are in all. Write its letter.

(A) Count with $6+6$ .
(B) Count with $6\times2$ .
(C) Count with $2\times6$ .
(D) Count by adding $6\times2$ twice.

The beads are arranged in $2$ rows with $6$ beads in each row. The round, colored beads form an array that is $6$ across and $2$ down, $12$ beads in all.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

There are 12 beads in an array of 2 rows with 6 beads in each row. Among four offered counting methods, we must find the one that does NOT give the correct total of 12.

Givens

The beads form an array 6 across and 2 down, 12 beads in all.
(A) 6+6; (B) 6 x 2; (C) 2 x 6; (D) add 6 x 2 twice.

Unknowns

Which lettered choice does not count to 12.

Constraints

A correct method must total exactly 12.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #2 Make a Systematic List

Picture the 6-by-2 array of beads, then evaluate each choice against the array and list the totals to spot the one that is not 12.

3 · Execute3 carry out the plan

1Read the array as the true total

#1 Draw a Diagram 3.OA.A.1

The beads are 2 rows of 6, so the correct total is 6 x 2 = 12. Any correct method must match this.

6 \times 2 = 12

An array of equal rows is exactly the product of rows and columns, so 2 rows of 6 is 12.

2Check the addition and product choices

#2 Make a Systematic List 2.OA.C.4

(A) 6+6 = 12, the 2 rows added. (B) 6 x 2 = 12 and (C) 2 x 6 = 12 are the same product in two orders. All three give 12.

6+6 = 12,\ 6\times2 = 12,\ 2\times6 = 12

Adding equal rows and multiplying rows by columns are just two views of the same array total.

3Check choice D

#2 Make a Systematic List 3.OA.A.1

Choice (D) adds 6 x 2 twice: 6 x 2 = 12, and 12 added 2 times is 24, which is not 12. So (D) is the incorrect method.

6 \times 2 = 12,\quad 12 \times 2 = 24 \neq 12

Each row holds 6 beads, not 6 x 2 = 12, so D double-counts every row and overshoots to 24.

Answer: D

4 · Reviewdoes it hold up?

Choices A, B, C all equal 12, the bead total, while D equals 24, twice as many, so D is clearly the wrong way to count.

Another way: You could split the array into halves: two rows give 6 x 2 = 12, so adding 6 x 2 only 1 times would rebuild the 12, showing that adding it twice is too many.

Standardsmin grade 3

2.OA.C.4 Use addition to find the total number of objects in rectangular arrays — Adding equal rows of 6 to total the bead array.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Reading 2 rows of 6 as the product 12 and judging each counting method.

💡Takeaway. Equal rows can be added or multiplied to the same total -- but counting each row as double overshoots, easy Grade 3 array sense!

Variant 2 easy

Find the choice that is not a correct way to count how many beads there are in all. Write its letter.

(A) Count with $8+8$ .
(B) Count with $8\times2$ .
(C) Count with $2\times8$ .
(D) Count by adding $8\times2$ twice.

The beads are arranged in $2$ rows with $8$ beads in each row. The round, colored beads form an array that is $8$ across and $2$ down, $16$ beads in all.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

There are 16 beads in an array of 2 rows with 8 beads in each row. Among four offered counting methods, we must find the one that does NOT give the correct total of 16.

Givens

The beads form an array 8 across and 2 down, 16 beads in all.
(A) 8+8; (B) 8 x 2; (C) 2 x 8; (D) add 8 x 2 twice.

Unknowns

Which lettered choice does not count to 16.

Constraints

A correct method must total exactly 16.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #2 Make a Systematic List

Picture the 8-by-2 array of beads, then evaluate each choice against the array and list the totals to spot the one that is not 16.

3 · Execute3 carry out the plan

1Read the array as the true total

#1 Draw a Diagram 3.OA.A.1

The beads are 2 rows of 8, so the correct total is 8 x 2 = 16. Any correct method must match this.

8 \times 2 = 16

An array of equal rows is exactly the product of rows and columns, so 2 rows of 8 is 16.

2Check the addition and product choices

#2 Make a Systematic List 2.OA.C.4

(A) 8+8 = 16, the 2 rows added. (B) 8 x 2 = 16 and (C) 2 x 8 = 16 are the same product in two orders. All three give 16.

8+8 = 16,\ 8\times2 = 16,\ 2\times8 = 16

Adding equal rows and multiplying rows by columns are just two views of the same array total.

3Check choice D

#2 Make a Systematic List 3.OA.A.1

Choice (D) adds 8 x 2 twice: 8 x 2 = 16, and 16 added 2 times is 32, which is not 16. So (D) is the incorrect method.

8 \times 2 = 16,\quad 16 \times 2 = 32 \neq 16

Each row holds 8 beads, not 8 x 2 = 16, so D double-counts every row and overshoots to 32.

Answer: D

4 · Reviewdoes it hold up?

Choices A, B, C all equal 16, the bead total, while D equals 32, twice as many, so D is clearly the wrong way to count.

Another way: You could split the array into halves: two rows give 8 x 2 = 16, so adding 8 x 2 only 1 times would rebuild the 16, showing that adding it twice is too many.

Standardsmin grade 3

2.OA.C.4 Use addition to find the total number of objects in rectangular arrays — Adding equal rows of 8 to total the bead array.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Reading 2 rows of 8 as the product 16 and judging each counting method.

💡Takeaway. Equal rows can be added or multiplied to the same total -- but counting each row as double overshoots, easy Grade 3 array sense!

Variant 3 easy

Find the choice that is not a correct way to count how many beads there are in all. Write its letter.

(A) Count with $5+5+5$ .
(B) Count with $5\times3$ .
(C) Count with $3\times5$ .
(D) Count by adding $5\times2$ three times.

The beads are arranged in $3$ rows with $5$ beads in each row. The round, colored beads form an array that is $5$ across and $3$ down, $15$ beads in all.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

There are 15 beads in an array of 3 rows with 5 beads in each row. Among four offered counting methods, we must find the one that does NOT give the correct total of 15.

Givens

The beads form an array 5 across and 3 down, 15 beads in all.
(A) 5+5+5; (B) 5 x 3; (C) 3 x 5; (D) add 5 x 2 three times.

Unknowns

Which lettered choice does not count to 15.

Constraints

A correct method must total exactly 15.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #2 Make a Systematic List

Picture the 5-by-3 array of beads, then evaluate each choice against the array and list the totals to spot the one that is not 15.

3 · Execute3 carry out the plan

1Read the array as the true total

#1 Draw a Diagram 3.OA.A.1

The beads are 3 rows of 5, so the correct total is 5 x 3 = 15. Any correct method must match this.

5 \times 3 = 15

An array of equal rows is exactly the product of rows and columns, so 3 rows of 5 is 15.

2Check the addition and product choices

#2 Make a Systematic List 2.OA.C.4

(A) 5+5+5 = 15, the 3 rows added. (B) 5 x 3 = 15 and (C) 3 x 5 = 15 are the same product in two orders. All three give 15.

5+5+5 = 15,\ 5\times3 = 15,\ 3\times5 = 15

Adding equal rows and multiplying rows by columns are just two views of the same array total.

3Check choice D

#2 Make a Systematic List 3.OA.A.1

Choice (D) adds 5 x 2 three times: 5 x 2 = 10, and 10 added 3 times is 30, which is not 15. So (D) is the incorrect method.

5 \times 2 = 10,\quad 10 \times 3 = 30 \neq 15

Each row holds 5 beads, not 5 x 2 = 10, so D double-counts every row and overshoots to 30.

Answer: D

4 · Reviewdoes it hold up?

Choices A, B, C all equal 15, the bead total, while D equals 30, twice as many, so D is clearly the wrong way to count.

Another way: You could split the array into halves: two rows give 5 x 2 = 10, so adding 5 x 2 only 1.5 times would rebuild the 15, showing that adding it three times is too many.

Standardsmin grade 3

2.OA.C.4 Use addition to find the total number of objects in rectangular arrays — Adding equal rows of 5 to total the bead array.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Reading 3 rows of 5 as the product 15 and judging each counting method.

💡Takeaway. Equal rows can be added or multiplied to the same total -- but counting each row as double overshoots, easy Grade 3 array sense!

Variant 4 medium

Find the choice that is not a correct way to count how many beads there are in all. Write its letter.

(A) Count with $7+7+7$ .
(B) Count with $7\times3$ .
(C) Count with $3\times7$ .
(D) Count by adding $7\times2$ three times.

The beads are arranged in $3$ rows with $7$ beads in each row. The round, colored beads form an array that is $7$ across and $3$ down, $21$ beads in all.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

There are 21 beads in an array of 3 rows with 7 beads in each row. Among four offered counting methods, we must find the one that does NOT give the correct total of 21.

Givens

The beads form an array 7 across and 3 down, 21 beads in all.
(A) 7+7+7; (B) 7 x 3; (C) 3 x 7; (D) add 7 x 2 three times.

Unknowns

Which lettered choice does not count to 21.

Constraints

A correct method must total exactly 21.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #2 Make a Systematic List

Picture the 7-by-3 array of beads, then evaluate each choice against the array and list the totals to spot the one that is not 21.

3 · Execute3 carry out the plan

1Read the array as the true total

#1 Draw a Diagram 3.OA.A.1

The beads are 3 rows of 7, so the correct total is 7 x 3 = 21. Any correct method must match this.

7 \times 3 = 21

An array of equal rows is exactly the product of rows and columns, so 3 rows of 7 is 21.

2Check the addition and product choices

#2 Make a Systematic List 2.OA.C.4

(A) 7+7+7 = 21, the 3 rows added. (B) 7 x 3 = 21 and (C) 3 x 7 = 21 are the same product in two orders. All three give 21.

7+7+7 = 21,\ 7\times3 = 21,\ 3\times7 = 21

Adding equal rows and multiplying rows by columns are just two views of the same array total.

3Check choice D

#2 Make a Systematic List 3.OA.A.1

Choice (D) adds 7 x 2 three times: 7 x 2 = 14, and 14 added 3 times is 42, which is not 21. So (D) is the incorrect method.

7 \times 2 = 14,\quad 14 \times 3 = 42 \neq 21

Each row holds 7 beads, not 7 x 2 = 14, so D double-counts every row and overshoots to 42.

Answer: D

4 · Reviewdoes it hold up?

Choices A, B, C all equal 21, the bead total, while D equals 42, twice as many, so D is clearly the wrong way to count.

Another way: You could split the array into halves: two rows give 7 x 2 = 14, so adding 7 x 2 only 1.5 times would rebuild the 21, showing that adding it three times is too many.

Standardsmin grade 3

2.OA.C.4 Use addition to find the total number of objects in rectangular arrays — Adding equal rows of 7 to total the bead array.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Reading 3 rows of 7 as the product 21 and judging each counting method.

💡Takeaway. Equal rows can be added or multiplied to the same total -- but counting each row as double overshoots, easy Grade 3 array sense!

Variant 5 medium

Find the choice that is not a correct way to count how many beads there are in all. Write its letter.

(A) Count with $7+7+7+7$ .
(B) Count with $7\times4$ .
(C) Count with $4\times7$ .
(D) Count by adding $7\times2$ four times.

The beads are arranged in $4$ rows with $7$ beads in each row. The round, colored beads form an array that is $7$ across and $4$ down, $28$ beads in all.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

There are 28 beads in an array of 4 rows with 7 beads in each row. Among four offered counting methods, we must find the one that does NOT give the correct total of 28.

Givens

The beads form an array 7 across and 4 down, 28 beads in all.
(A) 7+7+7+7; (B) 7 x 4; (C) 4 x 7; (D) add 7 x 2 four times.

Unknowns

Which lettered choice does not count to 28.

Constraints

A correct method must total exactly 28.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #2 Make a Systematic List

Picture the 7-by-4 array of beads, then evaluate each choice against the array and list the totals to spot the one that is not 28.

3 · Execute3 carry out the plan

1Read the array as the true total

#1 Draw a Diagram 3.OA.A.1

The beads are 4 rows of 7, so the correct total is 7 x 4 = 28. Any correct method must match this.

7 \times 4 = 28

An array of equal rows is exactly the product of rows and columns, so 4 rows of 7 is 28.

2Check the addition and product choices

#2 Make a Systematic List 2.OA.C.4

(A) 7+7+7+7 = 28, the 4 rows added. (B) 7 x 4 = 28 and (C) 4 x 7 = 28 are the same product in two orders. All three give 28.

7+7+7+7 = 28,\ 7\times4 = 28,\ 4\times7 = 28

Adding equal rows and multiplying rows by columns are just two views of the same array total.

3Check choice D

#2 Make a Systematic List 3.OA.A.1

Choice (D) adds 7 x 2 four times: 7 x 2 = 14, and 14 added 4 times is 56, which is not 28. So (D) is the incorrect method.

7 \times 2 = 14,\quad 14 \times 4 = 56 \neq 28

Each row holds 7 beads, not 7 x 2 = 14, so D double-counts every row and overshoots to 56.

Answer: D

4 · Reviewdoes it hold up?

Choices A, B, C all equal 28, the bead total, while D equals 56, twice as many, so D is clearly the wrong way to count.

Another way: You could split the array into halves: two rows give 7 x 2 = 14, so adding 7 x 2 only twice would rebuild the 28, showing that adding it four times is too many.

Standardsmin grade 3

2.OA.C.4 Use addition to find the total number of objects in rectangular arrays — Adding equal rows of 7 to total the bead array.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Reading 4 rows of 7 as the product 28 and judging each counting method.

💡Takeaway. Equal rows can be added or multiplied to the same total -- but counting each row as double overshoots, easy Grade 3 array sense!

Variant 6 medium

Find the choice that is not a correct way to count how many beads there are in all. Write its letter.

(A) Count with $8+8+8+8$ .
(B) Count with $8\times4$ .
(C) Count with $4\times8$ .
(D) Count by adding $8\times2$ four times.

The beads are arranged in $4$ rows with $8$ beads in each row. The round, colored beads form an array that is $8$ across and $4$ down, $32$ beads in all.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

There are 32 beads in an array of 4 rows with 8 beads in each row. Among four offered counting methods, we must find the one that does NOT give the correct total of 32.

Givens

The beads form an array 8 across and 4 down, 32 beads in all.
(A) 8+8+8+8; (B) 8 x 4; (C) 4 x 8; (D) add 8 x 2 four times.

Unknowns

Which lettered choice does not count to 32.

Constraints

A correct method must total exactly 32.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #2 Make a Systematic List

Picture the 8-by-4 array of beads, then evaluate each choice against the array and list the totals to spot the one that is not 32.

3 · Execute3 carry out the plan

1Read the array as the true total

#1 Draw a Diagram 3.OA.A.1

The beads are 4 rows of 8, so the correct total is 8 x 4 = 32. Any correct method must match this.

8 \times 4 = 32

An array of equal rows is exactly the product of rows and columns, so 4 rows of 8 is 32.

2Check the addition and product choices

#2 Make a Systematic List 2.OA.C.4

(A) 8+8+8+8 = 32, the 4 rows added. (B) 8 x 4 = 32 and (C) 4 x 8 = 32 are the same product in two orders. All three give 32.

8+8+8+8 = 32,\ 8\times4 = 32,\ 4\times8 = 32

Adding equal rows and multiplying rows by columns are just two views of the same array total.

3Check choice D

#2 Make a Systematic List 3.OA.A.1

Choice (D) adds 8 x 2 four times: 8 x 2 = 16, and 16 added 4 times is 64, which is not 32. So (D) is the incorrect method.

8 \times 2 = 16,\quad 16 \times 4 = 64 \neq 32

Each row holds 8 beads, not 8 x 2 = 16, so D double-counts every row and overshoots to 64.

Answer: D

4 · Reviewdoes it hold up?

Choices A, B, C all equal 32, the bead total, while D equals 64, twice as many, so D is clearly the wrong way to count.

Another way: You could split the array into halves: two rows give 8 x 2 = 16, so adding 8 x 2 only twice would rebuild the 32, showing that adding it four times is too many.

Standardsmin grade 3

2.OA.C.4 Use addition to find the total number of objects in rectangular arrays — Adding equal rows of 8 to total the bead array.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Reading 4 rows of 8 as the product 32 and judging each counting method.

💡Takeaway. Equal rows can be added or multiplied to the same total -- but counting each row as double overshoots, easy Grade 3 array sense!

Variant 7 medium

Find the choice that is not a correct way to count how many beads there are in all. Write its letter.

(A) Count with $4+4+4+4+4$ .
(B) Count with $4\times5$ .
(C) Count with $5\times4$ .
(D) Count by adding $4\times2$ five times.

The beads are arranged in $5$ rows with $4$ beads in each row. The round, colored beads form an array that is $4$ across and $5$ down, $20$ beads in all.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

There are 20 beads in an array of 5 rows with 4 beads in each row. Among four offered counting methods, we must find the one that does NOT give the correct total of 20.

Givens

The beads form an array 4 across and 5 down, 20 beads in all.
(A) 4+4+4+4+4; (B) 4 x 5; (C) 5 x 4; (D) add 4 x 2 five times.

Unknowns

Which lettered choice does not count to 20.

Constraints

A correct method must total exactly 20.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #2 Make a Systematic List

Picture the 4-by-5 array of beads, then evaluate each choice against the array and list the totals to spot the one that is not 20.

3 · Execute3 carry out the plan

1Read the array as the true total

#1 Draw a Diagram 3.OA.A.1

The beads are 5 rows of 4, so the correct total is 4 x 5 = 20. Any correct method must match this.

4 \times 5 = 20

An array of equal rows is exactly the product of rows and columns, so 5 rows of 4 is 20.

2Check the addition and product choices

#2 Make a Systematic List 2.OA.C.4

(A) 4+4+4+4+4 = 20, the 5 rows added. (B) 4 x 5 = 20 and (C) 5 x 4 = 20 are the same product in two orders. All three give 20.

4+4+4+4+4 = 20,\ 4\times5 = 20,\ 5\times4 = 20

Adding equal rows and multiplying rows by columns are just two views of the same array total.

3Check choice D

#2 Make a Systematic List 3.OA.A.1

Choice (D) adds 4 x 2 five times: 4 x 2 = 8, and 8 added 5 times is 40, which is not 20. So (D) is the incorrect method.

4 \times 2 = 8,\quad 8 \times 5 = 40 \neq 20

Each row holds 4 beads, not 4 x 2 = 8, so D double-counts every row and overshoots to 40.

Answer: D

4 · Reviewdoes it hold up?

Choices A, B, C all equal 20, the bead total, while D equals 40, twice as many, so D is clearly the wrong way to count.

Another way: You could split the array into halves: two rows give 4 x 2 = 8, so adding 4 x 2 only 2.5 times would rebuild the 20, showing that adding it five times is too many.

Standardsmin grade 3

2.OA.C.4 Use addition to find the total number of objects in rectangular arrays — Adding equal rows of 4 to total the bead array.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Reading 5 rows of 4 as the product 20 and judging each counting method.

💡Takeaway. Equal rows can be added or multiplied to the same total -- but counting each row as double overshoots, easy Grade 3 array sense!

Variant 8 hard

Find the choice that is not a correct way to count how many beads there are in all. Write its letter.

(A) Count with $6+6+6+6+6$ .
(B) Count with $6\times5$ .
(C) Count with $5\times6$ .
(D) Count by adding $6\times2$ five times.

The beads are arranged in $5$ rows with $6$ beads in each row. The round, colored beads form an array that is $6$ across and $5$ down, $30$ beads in all.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

There are 30 beads in an array of 5 rows with 6 beads in each row. Among four offered counting methods, we must find the one that does NOT give the correct total of 30.

Givens

The beads form an array 6 across and 5 down, 30 beads in all.
(A) 6+6+6+6+6; (B) 6 x 5; (C) 5 x 6; (D) add 6 x 2 five times.

Unknowns

Which lettered choice does not count to 30.

Constraints

A correct method must total exactly 30.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #2 Make a Systematic List

Picture the 6-by-5 array of beads, then evaluate each choice against the array and list the totals to spot the one that is not 30.

3 · Execute3 carry out the plan

1Read the array as the true total

#1 Draw a Diagram 3.OA.A.1

The beads are 5 rows of 6, so the correct total is 6 x 5 = 30. Any correct method must match this.

6 \times 5 = 30

An array of equal rows is exactly the product of rows and columns, so 5 rows of 6 is 30.

2Check the addition and product choices

#2 Make a Systematic List 2.OA.C.4

(A) 6+6+6+6+6 = 30, the 5 rows added. (B) 6 x 5 = 30 and (C) 5 x 6 = 30 are the same product in two orders. All three give 30.

6+6+6+6+6 = 30,\ 6\times5 = 30,\ 5\times6 = 30

Adding equal rows and multiplying rows by columns are just two views of the same array total.

3Check choice D

#2 Make a Systematic List 3.OA.A.1

Choice (D) adds 6 x 2 five times: 6 x 2 = 12, and 12 added 5 times is 60, which is not 30. So (D) is the incorrect method.

6 \times 2 = 12,\quad 12 \times 5 = 60 \neq 30

Each row holds 6 beads, not 6 x 2 = 12, so D double-counts every row and overshoots to 60.

Answer: D

4 · Reviewdoes it hold up?

Choices A, B, C all equal 30, the bead total, while D equals 60, twice as many, so D is clearly the wrong way to count.

Another way: You could split the array into halves: two rows give 6 x 2 = 12, so adding 6 x 2 only 2.5 times would rebuild the 30, showing that adding it five times is too many.

Standardsmin grade 3

2.OA.C.4 Use addition to find the total number of objects in rectangular arrays — Adding equal rows of 6 to total the bead array.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Reading 5 rows of 6 as the product 30 and judging each counting method.

💡Takeaway. Equal rows can be added or multiplied to the same total -- but counting each row as double overshoots, easy Grade 3 array sense!

Variant 9 hard

Find the choice that is not a correct way to count how many beads there are in all. Write its letter.

(A) Count with $3+3+3+3+3+3$ .
(B) Count with $3\times6$ .
(C) Count with $6\times3$ .
(D) Count by adding $3\times2$ six times.

The beads are arranged in $6$ rows with $3$ beads in each row. The round, colored beads form an array that is $3$ across and $6$ down, $18$ beads in all.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

There are 18 beads in an array of 6 rows with 3 beads in each row. Among four offered counting methods, we must find the one that does NOT give the correct total of 18.

Givens

The beads form an array 3 across and 6 down, 18 beads in all.
(A) 3+3+3+3+3+3; (B) 3 x 6; (C) 6 x 3; (D) add 3 x 2 six times.

Unknowns

Which lettered choice does not count to 18.

Constraints

A correct method must total exactly 18.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #2 Make a Systematic List

Picture the 3-by-6 array of beads, then evaluate each choice against the array and list the totals to spot the one that is not 18.

3 · Execute3 carry out the plan

1Read the array as the true total

#1 Draw a Diagram 3.OA.A.1

The beads are 6 rows of 3, so the correct total is 3 x 6 = 18. Any correct method must match this.

3 \times 6 = 18

An array of equal rows is exactly the product of rows and columns, so 6 rows of 3 is 18.

2Check the addition and product choices

#2 Make a Systematic List 2.OA.C.4

(A) 3+3+3+3+3+3 = 18, the 6 rows added. (B) 3 x 6 = 18 and (C) 6 x 3 = 18 are the same product in two orders. All three give 18.

3+3+3+3+3+3 = 18,\ 3\times6 = 18,\ 6\times3 = 18

Adding equal rows and multiplying rows by columns are just two views of the same array total.

3Check choice D

#2 Make a Systematic List 3.OA.A.1

Choice (D) adds 3 x 2 six times: 3 x 2 = 6, and 6 added 6 times is 36, which is not 18. So (D) is the incorrect method.

3 \times 2 = 6,\quad 6 \times 6 = 36 \neq 18

Each row holds 3 beads, not 3 x 2 = 6, so D double-counts every row and overshoots to 36.

Answer: D

4 · Reviewdoes it hold up?

Choices A, B, C all equal 18, the bead total, while D equals 36, twice as many, so D is clearly the wrong way to count.

Another way: You could split the array into halves: two rows give 3 x 2 = 6, so adding 3 x 2 only three times would rebuild the 18, showing that adding it six times is too many.

Standardsmin grade 3

2.OA.C.4 Use addition to find the total number of objects in rectangular arrays — Adding equal rows of 3 to total the bead array.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Reading 6 rows of 3 as the product 18 and judging each counting method.

💡Takeaway. Equal rows can be added or multiplied to the same total -- but counting each row as double overshoots, easy Grade 3 array sense!

Variant 10 hard

Find the choice that is not a correct way to count how many beads there are in all. Write its letter.

(A) Count with $4+4+4+4+4+4$ .
(B) Count with $4\times6$ .
(C) Count with $6\times4$ .
(D) Count by adding $4\times2$ six times.

The beads are arranged in $6$ rows with $4$ beads in each row. The round, colored beads form an array that is $4$ across and $6$ down, $24$ beads in all.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

There are 24 beads in an array of 6 rows with 4 beads in each row. Among four offered counting methods, we must find the one that does NOT give the correct total of 24.

Givens

The beads form an array 4 across and 6 down, 24 beads in all.
(A) 4+4+4+4+4+4; (B) 4 x 6; (C) 6 x 4; (D) add 4 x 2 six times.

Unknowns

Which lettered choice does not count to 24.

Constraints

A correct method must total exactly 24.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #2 Make a Systematic List

Picture the 4-by-6 array of beads, then evaluate each choice against the array and list the totals to spot the one that is not 24.

3 · Execute3 carry out the plan

1Read the array as the true total

#1 Draw a Diagram 3.OA.A.1

The beads are 6 rows of 4, so the correct total is 4 x 6 = 24. Any correct method must match this.

4 \times 6 = 24

An array of equal rows is exactly the product of rows and columns, so 6 rows of 4 is 24.

2Check the addition and product choices

#2 Make a Systematic List 2.OA.C.4

(A) 4+4+4+4+4+4 = 24, the 6 rows added. (B) 4 x 6 = 24 and (C) 6 x 4 = 24 are the same product in two orders. All three give 24.

4+4+4+4+4+4 = 24,\ 4\times6 = 24,\ 6\times4 = 24

Adding equal rows and multiplying rows by columns are just two views of the same array total.

3Check choice D

#2 Make a Systematic List 3.OA.A.1

Choice (D) adds 4 x 2 six times: 4 x 2 = 8, and 8 added 6 times is 48, which is not 24. So (D) is the incorrect method.

4 \times 2 = 8,\quad 8 \times 6 = 48 \neq 24

Each row holds 4 beads, not 4 x 2 = 8, so D double-counts every row and overshoots to 48.

Answer: D

4 · Reviewdoes it hold up?

Choices A, B, C all equal 24, the bead total, while D equals 48, twice as many, so D is clearly the wrong way to count.

Another way: You could split the array into halves: two rows give 4 x 2 = 8, so adding 4 x 2 only three times would rebuild the 24, showing that adding it six times is too many.

Standardsmin grade 3

2.OA.C.4 Use addition to find the total number of objects in rectangular arrays — Adding equal rows of 4 to total the bead array.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Reading 6 rows of 4 as the product 24 and judging each counting method.

💡Takeaway. Equal rows can be added or multiplied to the same total -- but counting each row as double overshoots, easy Grade 3 array sense!

Where next?

← Back to the problem: Count groups several equivalent ways