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What’s the Value? – Solve for Unknowns

Teacher Notes

To solve for values of unknowns in equations, students perform all computations following the order of operations: operations in parentheses first, followed by exponents, then multiplication and division from left to right, and then addition and subtraction from left to right. Operations within parentheses follow the standard order, as well. Encourage students to replace the variable with its value and check to be sure that the two expressions (to the left and to the right of the = symbol) name the same number.

Solutions:

1. \ \quad b + 2 \times 3 \times 2^2 \div 3 = 2(5 + 6) - 2\!\\{\;} \qquad b + 2 \times 3 \times 2^2 \div 3 = 2 \times 11 - 2\!\\{\;} \qquad b + 2 \times 3 \times 4 \div 3 = 2 \times 11 - 2\!\\{\;} \qquad b + 8 = 22 - 2\!\\{\;} \qquad b + 8 = 20\!\\{\;} \qquad b = 20 - 8\!\\{\;} \qquad b = 12

2. \ \quad 10^2 - 6(4 + 6) - 3^2 - 3 \times 4 - 1 = d + 50 \div 5^2\!\\{\;} \qquad 10^2 - 6 \times 10 - 3^2 - 3 \times 4 - 1 = d + 50 \div 5^2\!\\{\;} \qquad 100 - 6 \times 10 - 9 - 3 \times 4 - 1 = d + 50 \div 25\!\\{\;} \qquad 100 - 60 - 9 - 12 - 1 = d + 2\!\\{\;} \qquad 18 = d + 2\!\\{\;} \qquad 18 - 2 = d\!\\{\;} \qquad 16 = d

3. \ \quad 4(9 - 5) + h + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3\!\\{\;} \qquad 4 \times 4 + h + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3\!\\{\;} \qquad 4 \times 4 + h + 9 - 8 + 4 = 9 \times 3 - 2 \times 3\!\\{\;} \qquad 16 + h + 9 - 8 + 4 = 27 - 6\!\\{\;} \qquad h + 21 = 21\!\\{\;} \qquad h = 21 - 21\!\\{\;} \qquad h = 0

4. \ \quad 2 + 5^2 \div 5 \times 1 + 0 \times 34 = 2y - 7(4 - 3)\!\\{\;} \qquad 2 + 5^2 \div 5 \times 1 + 0 \times 34 = 2y - 7 \times 1\!\\{\;} \qquad 2 + 25 \div 5 \times 1 + 0 \times 34 = 2y - 7 \times 1\!\\{\;} \qquad 2 + 5 + 0 = 2y - 7\!\\{\;} \qquad 7 = 2y - 7\!\\{\;} \qquad 7 + 7 = 2y\!\\{\;} \qquad 14 = 2y\!\\{\;} \qquad 7 = y

What’s the Value? – Solve for Unknowns

6 \times 3^2 \div 2 + z + 4(7 - 3) + 2^3 \div 4 = 3^2 \times 6 + 1

What is the value of z?

& \mathbf{Describe:} && \text{The equation has parentheses, exponents, multiplication, division, and addition.}\\&&& z \ \text{is the variable.}\\\\& \mathbf{My \ Job:} && \text{Apply the order of operations rule to figure out the value of}\ z.\\\\& \mathbf{Solve:} && 6 \times 3^2 \div 2 + z + 4(7 - 3) + 2^3 \div 4 = 3^2 \times 6 + 1\\&&& \mathbf{Parenthesis} \qquad \qquad \quad 6 \times 3^2 \div 2 + z + 4 \times 4 + 2^3 \div 4 = 3^2 \times 6 + 1\\&&& \mathbf{Exponents} \qquad \qquad \quad \ 6 \times 9 \div 2 + z + 4 \times 4 + 8 \div 4 = 9 \times 6 + 1\\&&& \mathbf{Multiplication/} \qquad \quad 27 + z + 16 + 2 = 54 + 1\\&&& \mathbf{Division}\\&&& \mathbf{(left \ to \ right)}\\&&& \mathbf{Addition} \qquad \qquad \qquad \ 45 + z = 55\\&&& \mathbf{(left \  to \ right)} \qquad \qquad \ z = 55 - 45\\&&& \qquad \qquad \qquad \qquad \qquad \quad z = 10\\\\& \mathbf{Check:} && \text{Replace} \ z \ \text{with 10 in the equation. Check that the two expressions}\\&&&\text{(to the right and to the left of the = symbol) name the same number.}\\&&& 6 \times 3^2 \div 2 + 10 + 4(7 - 3) + 2^3 \div 4 = 3^2 \times 6 + 1\\&&& 6 \times 3^2 \div 2 + 10 + 4 \times 4 + 2^3 \div 4 = 3^2 \times 6 + 1\\&&& 6 \times 9 \div 2 + 10 + 4 \times 4 + 8 \div 4 = 9 \times 6 + 1\\&&& 27 + 10 + 16 + 2 = 54 + 1\\&&& 55 = 55

Follow the order of operations. Show each step.

What is the value of the variable?

1. \quad b + 2 \times 3 \times 2^2 \div 3 = 2(5 + 6) - 2\\\\\\\\\\{\;} \qquad b = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\\\2. \quad 10^2 - 6(4 + 6) - 3^2 - 3 \times 4 - 1 = d + 50 \div 5^2\\\\\\\\\\{\;} \qquad d = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\\\3. \quad 4(9 - 5) + h + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3\\\\\\\\\\{\;} \qquad h = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\\\4. \quad 2 + 5^2 \div 5 \times 1 + 0 \times 34 = 2y - 7(4 - 3)\\\\\\\\\\{\;} \qquad y = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}

Extra for Experts: What’s the Value? – Solve for Unknowns

Follow the order of operations. Show each step.

What is the value of the variable?

1. \quad 2a + 5(9 - 8) \times 2^2 = 2^2 \times 3^2 + 2\\\\\\\\\\{\;} \qquad a = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\\\2. \quad 9^2 - 8^2 - 16 + 4^3 + 2^2 = e(5 + 2) - 1\\\\\\\\\\{\;} \qquad e = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\\\3. \quad 2m + 3 \times 9 \div 3^3 + 4(8 - 3) = 7^2 - 3 \times 6\\\\\\\\\\{\;} \qquad m = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\\\4. \quad 7^2 \div 7 \times (5^2 - 17) - (2 \times 6) = 4l - 2^2 \times 5\\\\\\\\\\{\;} \qquad l = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}

Solutions:

1. \ \quad 2a + 5(9 - 8) \times 2^2 = 2^2 \times 3^2 + 2\!\\{\;} \quad \quad 2a + 5 \times 1 \times 2^2 = 2^2 \times 3^2 + 2\!\\{\;} \quad \quad 2a + 5 \times 1 \times 4 = 4 \times 9 + 2\!\\{\;} \quad \quad 2a + 20 = 36 + 2\!\\{\;} \quad \quad 2a + 20 = 38\!\\{\;} \quad \quad 2a = 38 - 20\!\\{\;} \quad \quad 2a = 18\!\\{\;} \quad \quad a = 9\!\\\\2. \quad \ 9^2 - 8^2 - 16 + 4^3 + 2^2 = e(5 + 2) - 1\!\\{\;} \quad \quad 9^2 - 8^2 - 16 + 4^3 + 2^2 = e \times 7 - 1\!\\{\;} \quad \quad 81 - 64 - 16 + 64 + 4= 7e - 1\!\\{\;} \quad \quad 69 = 7e - 1\!\\{\;} \quad \quad 69 + 1 = 7e\!\\{\;} \quad \quad 70 = 7e\!\\{\;} \quad \quad 10 = e\!\\\\3. \quad \ 2m + 3 \times 9 \div 3^3 + 4(8 - 3) = 7^2 - 3 \times 6\!\\{\;} \quad \quad 2m + 3 \times 9 \div 3^3 + 4 \times 5 = 7^2 - 3 \times 6\!\\{\;} \quad \quad 2m + 3 \times 9 \div 27 + 4 \times 5 = 49 - 3 \times 6\!\\{\;} \quad \quad 2m + 1 + 20 = 49 - 18\!\\{\;} \quad \quad 2m + 21 = 31\!\\{\;} \quad \quad 2m = 31 - 21\!\\{\;} \quad \quad 2m = 10\!\\{\;} \quad \quad m = 5\!\\\\4. \quad \ 7^2 \div 7 \times (5^2 - 17) - (2 \times 6) = 4l - 2^2 \times 5\!\\{\;} \quad \quad 7^2 \div 7 \times (25 - 17) - (2 \times 6) = 4l - 2^2 \times 5\!\\{\;} \quad \quad 7^2 \div 7 \times 8 - 12 = 4l - 2^2 \times 5\!\\{\;} \quad \quad 49 \div 7 \times 8 - 12 = 4l - 4 \times 5\!\\{\;} \quad \quad 56 - 12 = 4l - 20\!\\{\;} \quad \quad 44 = 4l - 20\!\\{\;} \quad \quad 44 + 20 = 4l\!\\{\;} \quad \quad 64 = 4l\!\\{\;} \quad \quad 16 = l

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