To solve this "Algebra Calculation Game," we need to simplify each algebraic expression found in the hexagons. The goal is usually to expand brackets (remove parentheses) and combine like terms so that each expression is in its simplest form.
Here is the step-by-step simplification for every expression on the board:
Row 1 (Top)
1.
$n^3$: This is already simplified.
2.
$6n - 2$: This is already simplified.
Row 2
3.
$3n - 5$: Already simplified.
4.
$2n - 5$: Already simplified.
5.
$3(n - 1)$:
* Multiply $3$ by $n$: $3n$
* Multiply $3$ by $-1$: $-3$
* Result:
$3n - 3$
Row 3
6.
$10 - 3n$: Already simplified.
7.
$\frac{1}{2}(n + 1)$:
* Multiply $\frac{1}{2}$ by $n$: $\frac{n}{2}$ (or $0.5n$)
* Multiply $\frac{1}{2}$ by $1$: $\frac{1}{2}$ (or $0.5$)
* Result:
$\frac{n}{2} + \frac{1}{2}$ (or $0.5n + 0.5$)
8.
$n^2 - 3$: Already simplified.
Row 4
9.
$2n^2$: Already simplified.
10.
$n^2 + 6$: Already simplified.
11.
$3(n + 2)$:
* Multiply $3$ by $n$: $3n$
* Multiply $3$ by $2$: $6$
* Result:
$3n + 6$
12.
$2(n - 3)$:
* Multiply $2$ by $n$: $2n$
* Multiply $2$ by $-3$: $-6$
* Result:
$2n - 6$
Row 5
13.
$10 - n^2$: Already simplified.
14.
$n(n + 2)$:
* Multiply $n$ by $n$: $n^2$
* Multiply $n$ by $2$: $2n$
* Result:
$n^2 + 2n$
15.
$5 - 2n$: Already simplified.
Row 6
16.
$(n + 3)^2$:
* This means $(n + 3)(n + 3)$.
* $n \times n = n^2$
* $n \times 3 = 3n$
* $3 \times n = 3n$
* $3 \times 3 = 9$
* Combine like terms ($3n + 3n = 6n$):
* Result:
$n^2 + 6n + 9$
17.
$n^2 - 2n$: Already simplified.
Row 7 (Bottom)
18.
$6n - n^2$: Already simplified.
19.
$2(2n - 1)$:
* Multiply $2$ by $2n$: $4n$
* Multiply $2$ by $-1$: $-2$
* Result:
$4n - 2$
20.
$(n - 2)^2$:
* This means $(n - 2)(n - 2)$.
* $n \times n = n^2$
* $n \times -2 = -2n$
* $-2 \times n = -2n$
* $-2 \times -2 = +4$
* Combine like terms ($-2n - 2n = -4n$):
* Result:
$n^2 - 4n + 4$
***
Summary of Simplified Expressions
If you are looking for matching pairs or specific values, here are the expressions that required calculation:
* $3(n-1) = \mathbf{3n - 3}$
* $\frac{1}{2}(n+1) = \mathbf{0.5n + 0.5}$
* $3(n+2) = \mathbf{3n + 6}$
* $2(n-3) = \mathbf{2n - 6}$
* $n(n+2) = \mathbf{n^2 + 2n}$
* $(n+3)^2 = \mathbf{n^2 + 6n + 9}$
* $2(2n-1) = \mathbf{4n - 2}$
* $(n-2)^2 = \mathbf{n^2 - 4n + 4}$
All other expressions on the board ($n^3$, $6n-2$, $3n-5$, $2n-5$, $10-3n$, $n^2-3$, $2n^2$, $n^2+6$, $10-n^2$, $5-2n$, $n^2-2n$, $6n-n^2$) are already in their simplest forms.
Final Answer:
The simplified forms for the expressions requiring expansion are:
1.
$3(n-1) \rightarrow 3n - 3$
2.
$\frac{1}{2}(n+1) \rightarrow \frac{n}{2} + \frac{1}{2}$
3.
$3(n+2) \rightarrow 3n + 6$
4.
$2(n-3) \rightarrow 2n - 6$
5.
$n(n+2) \rightarrow n^2 + 2n$
6.
$(n+3)^2 \rightarrow n^2 + 6n + 9$
7.
$2(2n-1) \rightarrow 4n - 2$
8.
$(n-2)^2 \rightarrow n^2 - 4n + 4$
Parent Tip: Review the logic above to help your child master the concept of algebra 1 fun worksheet.
