Grade 5 – Operations and Algebraic Thinking SOLUTIONS
1. A chocolate box has 3 trays of chocolate with 12 chocolates in each tray. Write an
equation to represent the total number of chocolates in the box. This is the number of
trays times the number of chocolates in each tray. Each box has 3 trays, each with 12
chocolates, so in total we have 3x12= 3x(2x6)=36.
2. What numbers can you make with 1, 2, 3, and 4? Using the operations of addition,
subtraction, and multiplication, we can make many different numbers. For example, we can
write 16 as
16 = (3×5) + 1.
You can use parentheses as many times as you like but each of the numbers 1, 2, 3, and 4 can
be used at most once. Find as many ways to make 24 and 16.
It would seem as if 32 is the largest that can be produced.
1=1
2=2
3=3
4=4
4+1=5
4+2=6
4+3=7
4+3+1=8
4+3+2=9
1+2+3+4=10
4x3-1=11
4x3=12
4x3+1=13
4x3+2=14
4x3+2+1=15
4x(3+1)=16
3x(4+1)+2=17
3x(4-1)x2=18
4x(3+2)-1=19
4x(3+2)=20
4x(3+2)+1=21
2x(4x3-1)=22
2x3x4-1=23
2x3x4=24
2x3x4+1=25
2x(4x3+1)=26
(4x2+1)x3=27
28?
29?
(4+1)x3x2=30
31?
4x(3+1)x2=32
33?
34?
35?
Besides the above we also have
 4x(3+2-1)=16
 4x2x(3-1)=16
 4x(3+2+1)=24
Note that we are not distinguishing different orders of the same basic expression. (i.e. we do not
view 2x3x4=24 as different from 2x4x3=24.)
3. True of False Explain your reasoning for the following
2 x (9 +7) = (9 +7) + ( 9 +7)
This is true since we are taking two of these things called 9+7. When you have two of something
you can think of that as 1 something + 1 something=2 somethings.
4. True of False Explain your reasoning for the following problems
67 + 86 = 68 + 85
From the left side to the right side the 67 goes up 1 to 68 and the 86 goes down one
to 85 so the two sides are in balance and this is true!
5. Show how to solve simply: 43+ ___= 48 + 76
Comparing the two sides of the equation we see that if 43 is increased by 5 to become 48 then the
76 must be 5 less than the blank so this must be 81.
6. Solve for m
m+m+m = m+12
Comparing the left and right sides we see that one of the ms matches. This means
that the other 2 ms must match the 12 so 2m=12 so m=6.
7. Insert parenthesis to make the following statement true
4 x 7 + 3 = 31
No parenthesis are needed since multiplication comes first so that this
equals (4x7)+3=31.
8. If you could insert one pair of parenthesis, which is the largest number you can make
6 x 5 ÷ 12-3
If we want a large answer we should divide by the smallest possible value. Thus we will make
our divisor 12-3=9 and hence 6x5÷(12-3)=30÷9=3 1/3.
9. Solve the following problems using the Order of Operations
60 +15÷3-6
=60+5-6=65-6=59
10. Solve the following problems using the Order of Operations
4x5+3–2÷2
=20+3-1=23-1=22.
11.
With 3 tables we have 11 seats.
Notice that adding an additional table brings in 3 new seats (one seat gets moved from it’s place on the
middle table to the one on the end). It also happens that going from 1 to two tables increases the
number of seats by 3 (going from 5 to 8). You can in fact see that each table gives 3 seats and then there
are 2 more (one on each end). Thus with n tables we will have 3n+2 seats in total.