Two-Step Problems and Patterns: practice solutions, Grade 3 – download the PDF

1 Parentheses matter ★★★

a) Multiply first: \(3\times 4=12\), then \(5+12=17\).

b) Parentheses first: \(5+3=8\), then \(8\times 4=32\).

The two answers are different (17 and 32) because the parentheses change the order of the steps.

2 Stickers ★★★

Step one: \(3\times 6=18\). Step two: \(18+4=22\).

Mia has 22 stickers.

3 Find the unknown ★★★

a) Undo the addition: \(n=45-18=27\). Check: \(27+18=45\).

b) Undo the multiplication: \(m=56\div 7=8\). Check: \(7\times 8=56\).

4 Even or odd? ★★★

Look at the last digit. Even (last digit 0, 2, 4, 6 or 8): 14, 60, 148. Odd (last digit 1, 3, 5, 7 or 9): 27, 83, 105.

5 Addition table row ★★★

a) 9, 10, 11, 12.

b) Each time you move one cell to the right, the sum goes up by 1.

c) \(8+5=12+1=13\).

6 Multiples of 5 ★★★

5, 10, 15, 20, 25, 30, 35, 40.

Multiples of 5 end in 0 or 5. 65 ends in 5, so it can be a multiple of 5 (\(5\times 13=65\)). 72 ends in 2, so it is not.

7 Quick estimate ★★★

58 rounds to 60 and 34 rounds to 30, so the estimate is \(60+30=90\).

Exact sum: \(58+34=92\). It is close to 90, so it is reasonable.

8 Muffin trays ★★★

Step one: \(6\times 8=48\) muffins. Step two: \(48-19=29\).

There are 29 muffins left.

9 Notebooks and a pen ★★★

a) \(t=4\times 3+2\).

b) \(4\times 3=12\), then \(12+2=14\). So \(t=14\) dollars.

c) \(20-14=6\). Leo gets 6 dollars in change.

10 Two unknowns ★★★

a) \(n=42\div 6=7\). Check: \(7\times 6=42\).

b) Undo the subtraction by adding: \(n=29+15=44\). Check: \(44-15=29\).

11 Order of operations ★★★

a) \(36\div 6=6\) and \(4\times 2=8\), so \(6+8=14\).

b) \(2\times 7=14\), so \(20-14=6\).

c) Parentheses first: \(2+4=6\), then \(36\div 6=6\).

d) \(20-2=18\), then \(18\times 7=126\).

12 Is it reasonable? ★★★

Round: \(50+40=90\). Sam’s answer 77 is much smaller than 90, so it is not reasonable.

Correct sum: \(49+38=87\), which is close to 90.

13 Facts from facts ★★★

a) The table is symmetric, so \(7\times 6=42\).

b) \(6\times 8\) is one more group of 6: \(42+6=48\).

c) \(6\times 5=30\) and \(6\times 2=12\), and \(30+12=42\).

14 Even or odd sums ★★★

a) even + even = even. \(24+36=60\), even.

b) odd + odd = even. \(17+25=42\), even.

c) even + odd = odd. \(18+31=49\), odd.

d) \(9+9=18\) is even, and even + odd (9) is odd. \(9+9+9=27\), odd.

15 Reading marathon ★★★

a) \(p=8\times 5+13\). \(8\times 5=40\) and \(40+13=53\). So \(p=53\) pages.

b) \(90-53=37\). Ava still has 37 pages to read.

16 Two-step equations ★★★

a) Undo the addition: \(3\times n=29-5=24\). Undo the multiplication: \(n=24\div 3=8\). Check: \(3\times 8+5=29\).

b) \(4\times n\) must be what is taken away from 50 to get 26: \(4\times n=50-26=24\), so \(n=24\div 4=6\). Check: \(50-4\times 6=50-24=26\).

17 Pencil boxes ★★★

a) Round 28 to 30: \(4\times 30=120\). The estimate is about 120, so 212 is far too big and is not reasonable.

b) \(4\times 20=80\) and \(4\times 8=32\), so \(80+32=112\). There are 112 pencils, close to the estimate 120.

18 Addition table hunt ★★★

a) \(1+6, 2+5, 3+4, 4+3, 5+2, 6+1\): 6 cells.

b) Only \(6+6\): 1 cell.

c) \(5+6\) and \(6+5\): 2 cells.

d) \(1+5, 2+4, 3+3, 4+2, 5+1\): 5 cells.

Each sum fills one diagonal. The middle diagonal (sum 7) is the longest, and the diagonals get shorter toward the corners.

19 Nines trick ★★★

a) \(9\times 6\): tens digit 5, ones digit \(9-5=4\), so 54. \(9\times 7\): tens digit 6, ones digit 3, so 63.

b) Pattern: tens digit 7, ones digit 2, so 72. Other method: \(9\times 8=10\times 8-8=80-8=72\). Both agree.

c) \(2\times 7=14\), and \(4\times 7\) is double that: \(14+14=28\).

20 Even and odd reasoning ★★★

a) \(3\times 7=21\) odd. \(4\times 9=36\) even. \(5\times 6=30\) even. \(7\times 7=49\) odd. A product is even as soon as one factor is even.

b) False. \(11+13=24\) is even, and \(24+15=39\) is odd. One example is enough to show the statement is false.

c) No, 25 is odd. There are 12 pairs and 1 student left over.

21 Family tickets ★★★

a) \(T=3\times 5+2\times 8\).

b) \(3\times 5=15\) and \(2\times 8=16\), so \(T=15+16=31\) dollars.

c) They pay 40 dollars: \(40-31=9\). They get 9 dollars in change.