solving problems using equations 1 - wordpress.com · 500 20 25 it will take 25 weeks to save $550....

OPEN-ENDEDPathway 1

Becca has created an equation to model a problem. She can solve the problem by solving the equation or graphing it.

I had $5 in my savings account and then deposited $10 each week. I now have $165. How many weeks have I been saving?

10w 1 5 5 165

• Create 3 problems that can be represented using an equation of either form:

❚p 1 ❚ 5 ❚ ❚k 2 ❚ 5 ❚

• Solve each problem by solving the equation or graphing it. If you use a graph, create it on grid paper.

• Describe or show how you solved each problem.

Equation 1: ____________________

problem:

Solution:

Solving Problems Using Equations

• You can use any letter to represent a variable.

• You can use a shortcut to show multiplication. e.g., 3n 5 3 3 n

• When you solve an equation, you determine the value of the variable that makes it true.

e.g., If 3n 5 6, n 5 2.

• You can solve an equation by thinking of a balance or a length model. e.g., 10w 1 5 5 65

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165510w

• To graph equations in the form ❚p 1 ❚ 5 ❚ and ❚k 2 ❚ 5 ❚, create a table of values for 3 ordered pairs and plot them. Then extend the points in a straight line or draw a line through them.

Remember

You will need• 1 cm Grid paper

(BLM 12) or 0.5 cm Grid paper (BLM 13)

Copyright © 2012 by Nelson Education Ltd.210

Leaps and BoundsSolving problems Using Equations, pathway 1

Leap SR 7-8_Topic 12.indd 210 2/25/12 4:14 PM

Equation 2: ____________________

problem:

Solution:

Equation 3: ____________________

problem:

Solution:


Leaps and Bounds Solving problems Using Equations, pathway 1


GuiDEDPathway 1

Ellen had $50 in the bank. Then she deposited $20 each week. How much money will she have in 15 weeks?How long will it take to save $550?

There are different ways to solve these problems.

Solving Equations Algebraically

• You can write an equation for each problem and then solve the equation.

The equation 50 1 (15 3 20) 5 m represents the amount of money Ellen will have in 15 weeks.

To solve it, calculate the left side.

50 1 15 3 20 5 m50 1 300 5 m

350 5 m

Ellen will have $350 in 15 weeks.

The equation 50 1 20w 5 550 represents how long it will take to save $550 because Ellen started with $50 and then added $20 each week.

– You can solve it algebraically by getting the variable by itself on one side of the equation.

50 1 20w 5 550 20w 5 500 Subtract 50 from each side.

w 5 25 Divide each side by 20.

It will take 25 weeks to save $550.

– You can also use a length model to represent the equation and then use the length model to solve the equation algebraically.

Model the equation.50 1 20w 5 550

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50

550

20w

Subtract 50 from both lengths. 20w 5 500

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50

550 � 50 � 500

20w

Divide both lengths by 20. w 5 25

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wwwwwwwwwwwwwwwwwwww

500 20 25


Solving Problems Using Equations


• You can use a shortcut to show multiplication involving a variable. e.g., 3n 5 3 3 n


e.g., If 3n 5 6, n 5 2.

Remember

You will need• 1 cm Grid paper

(BLM 12) or 0.5 cm Grid paper (BLM 13)


Solving problems Using Equations, pathway 1 Leaps and Bounds


Solving Equations by Graphing

• You can write an equation that relates the number of weeks (w )to the amount saved (s).

50 1 20w 5 s

You can use the equation to make a table of values, and then use the table to create a graph.

You can plot just 3 points and then extend them along a line, since the relationship is linear.

Now you can use the graph to solve both problems.

• What 2 equations would you write if Ellen deposited only $10 a week? How would you solve each equation?

Try TheseIf you use a graph to solve a problem, draw your graph on grid paper.

1. Which equation represents each problem below? Explain your thinking for each.

200 2 5w 5 50 200 2 50 2 5w 5 5 50 1 5w 5 200

a) Lisa had $50 in the bank. Then she deposited $5 a week. After how many weeks will she have $200?

b) Annie had $200 in the bank before she took out $50. Then she took out $5 every week. After how many weeks did she have only $5 left?

c) Lance had $200 in the bank before he took out $5 a week. After how many weeks did he have only $50 left?

Ellen’s Savings

Number of weeks

w

Amount saved ($)

s

0 50

5 150

10 250

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Number of weeks

Am

oun

t sa

ved

($)

50100150200250300350400450500550600

50 10 15 20 25 30

In 15 weeks,Ellen will havesaved $350.


Ellen’s Savings


Solving problems Using Equations, pathway 1Leaps and Bounds


2. Raj has read 40 pages of his book and plans to read 25 pages more each night. Use an equation to represent the problem. Then solve the problem. Show or describe your work.

a) How many pages will he have read after 10 more nights?

equation: _____________________

b) How long will it take Raj to read 690 pages?

equation: _____________________

3. Use an equation to solve each problem. Show or describe your work.

a) A rectangle that is 30 cm long has a perimeter of 92 cm. What is the width?

equation: _____________________

b) A rectangle is 30 cm long and 10 cm wide. What is the perimeter?

equation: _____________________

4. Use an equation to solve the problem below. Show or describe your work.

Alissa does yard work. She charges $18 to mow a lawn. Before she started her business, she spent $90 to repair her lawnmower. How many lawns would she have to mow to earn a profit of $450?

equation: _____________________


Solving problems Using Equations, pathway 1 Leaps and Bounds


5. Use an equation to solve the problem below.

For a special dinner, there were 221 people sitting at tables of 8 and one head table of 13. How many tables of 8 were there?

equation: _____________________

6. a) Change the problem in Question 5 so that it can be represented using the equation 320 5 6t 1 20.

b) Explain the changes you made and why.

7. Create and solve a problem for each equation.

a) 4m 5 84 b) 2m 2 30 5 18

What you have learned in this lesson about solving equations will help you when you are working with any equation.

FYI


Solving problems Using Equations, pathway 1Leaps and Bounds


OPEN-ENDEDPathway 2

Abi is using a balance model to represent the equation 3k 1 2 5 20.

Mila is representing the same equation, 3k 1 2 5 20, using a length model.

• Explain how each model above represents 3k 1 2 5 20.

balance model: length model:

• Use either model to solve for the value of k. Explain your strategy.

Solving Simple Equations

You will need• pan balance,

paper bags, and base ten blocks or linking cubes



Remember

• When you solve an equation, you determine the value of the variable that makes it true.e.g., If 3n 5 6, n 5 2.

Remember


Leaps and BoundsSolving Simple Equations, pathway 2


• Write and model 4 equations that use the variable k.– One or more equations should involve subtraction.– Three or more equations should involve multiplying k

by a number greater than 1.

• Sketch your models.

Equation 1: _____________________ Equation 3: _____________________

Equation 2: _____________________ Equation 4: _____________________

• Use your models to solve the equations.

Equation 1: k 5 ______________ Equation 3: k 5 ______________

Equation 2: k 5 ______________ Equation 4: k 5 ______________

• Explain how you solved 2 of the equations above.

Equation ____: _____________________ Equation ____: _____________________


Leaps and Bounds Solving Simple Equations, pathway 2


GuiDEDPathway 2

You will need• pan balance,

paper bags, and base ten cubes or linking cubes

Jorge is modelling 2 equations.An equation is a statement about 2 amounts being equal. The amounts can be numbers, for example, 2 1 4 5 5 1 1,or variables representing numbers, for example, 2m 1 3 5 11.

Jorge has used a length model and a balance model to represent the 2 equations.

An equation that has a variable can be solved in different ways.

using a Length Model

• You can use a length model to solve an equation.

You can think of the 2 sides of the equation as 2 matching rectangles, one above the other.

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m m 3

11

2m 3 11

To solve the equation 2m 1 3 5 11, you take away a length of 3 from each rectangle and the new rectangles still match in length.

The sections labelled m are equal, so 2m 5 8, and m 5 4.

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m

11 3 8

m

2m 8, so m 4.

2m 3 3 11 3

m m 3

11

Solving Simple Equations




e.g., If 3n 5 6, n 5 2.

Remember




using a Balance Model

• You can use a balance model to solve an equation.

You can think of the equation as 2 sides with the same mass.

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m m

2m 3 11

To solve the equation 2m 1 3 5 11, you can take 3 cubes from each side without changing the balance. That means each bag labelled m has 4 cubes in it, so m 5 4.

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m m m m

2m 8, so m 42m 3 3 11 3

Guessing and Testing

• You can also solve an equation by guessing and testing.

For example, to solve the equation 2 j 2 8 5 10, you can try different values for j in the equation that make sense, until you find a value that works.

Guess Test

first guess: j 5 6 2 3 6 2 8 5 4 too low

second guess: j 5 10 2 3 10 2 8 5 12 too high

third guess: j 5 9 2 3 9 2 8 5 10 j 5 9

Try These 1. What equation does each balance model represent?

a)

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x

_______________________

b)

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x x

_______________________

• If you do not have the materials to create a balance model, a simple sketch of the model will work.

Remember




2. What equation does each length model represent?

a)

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10

1x xx

_______________________

b)

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10

2xxxx

_______________________

3. Sketch a balance or length model to represent each equation.Do not solve it.

a) j 1 2 5 10 c) 2 j 1 2 5 10

b) k 1 3 5 11 d) 3 j 1 4 5 13

4. Use a model to help explain why each statement is true.

a) If 3x 1 5 5 11, then 3x 5 11 2 5. b) If 2x 5 14, then x 5 7.

5. You can add 1 to each side of the equation 2x 2 1 5 11 without changing the value of x. Use this length model to show or explain why this is possible.

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11

1

xx




6. Show or explain how you know that the equations m 1 3 5 8 and m 5 8 2 3 say the same thing.

7. What is the resulting equation each time?

a) Add 4 to both sides of 2m 2 4 5 8. ________________

b) Add 3 to both sides of 2m 2 3 5 9. ________________

c) Subtract 1 from both sides of 2m 1 1 5 27. ________________

d) Subtract 2 from both sides of 5m 1 2 5 18. ________________

8. Solve each equation. Show or explain your work.

a) 51 5 5q 1 6 b) 6m 2 4 5 68

9. Solve each equation.

a) g 1 10 5 15 g 5 ________

b) s 2 10 5 15 s 5 ________

c) 2f 5 40 f 5 ________

d) 3g 1 1 5 25 g 5 ________

e) 17 5 2p 1 9 p 5 ________

f ) 18 5 5e 2 2 e 5 ________

10. Does Jorge’s strategy make sense? Explain your thinking.

11. An equation has a variable with a value greater than 30. What could the equation be? List 3 possibilities.

______________________

______________________

______________________

Using models will be helpful even when you are solving equations that have solutions that are not whole numbers.

FYI

To solve 2p 2 3 5 11, first add 3 to both sides.




OPEN-ENDEDPathway 3

Renée is modelling the algebraic equation 2 1 3v 5 11 with algebra tiles. The equation means, “What number, when tripled and added to 2, results in 11?”

In the equation 2 1 3v 5 11, the variable v represents a single unknown amount. When you solve the equation, v 5 3.

Variables are also used to create equations that describe relationships among quantities. For example, the equation j 5 3 1 2w means,“The value of the variable j is 3 more than double the value of the variable w.”

In an equation that represents a relationship, there are multiple possible values for the variables that make it true. For example, in j 5 3 1 2w, if w 5 1, then j 5 5. If w 5 2, then j 5 6.

The equals sign in an equation means the algebraic expressions on both sides of the equation have the same value.

In 2 1 3v 5 11, 2 1 3v has the same value as 11.

In j 5 2 1 2w, j has the same value as 2 1 2w.

• Write an equation that uses a variable to represent a single unknown amount. Describe or show what your equation means. Solve the equation to determine what amount the variable represents.

equation: _______________________

Using Variables

algebraic equationa statement that uses an equals sign to show that 2 quantities have the same value

algebraic expressiona phrase that involves a set of numbers, operation signs, and variables

• Any letter can be used to represent a variable without changing the meaning of the expression or equation.

• You can use a shortcut to show multiplication involving a variable. e.g., 3n 5 3 3 n.

• When you solve an equation, you determine the value of the variable that makes the equation true. e.g.,

If 3n 5 6, then n 5 2. If j 5 2w and w 5 1,

then j 5 2. If w 5 2,then j 5 4, and so on.

Remember

You will need• algebra tiles: x-tiles

and 1-tiles


Leaps and BoundsUsing Variables, pathway 3


• Change one thing in your equation to make a new equation. Describe what your new equation means. Solve the equation.

new equation:

• Change something else in your equation to make a new equation. Describe what your new equation means. Solve the equation.

new equation:

• Write an equation that uses variables and represents a relationship. Describe what your equation means. Solve the equation to determine 2 sets of possible values for the variables.

equation: _______________________

• Change one thing in your equation. Describe what your new equation means. Solve the equation to determine 2 sets of possible values for the variables.

new equation:

• Change something else in your equation. Describe what your new equation means. Solve the equation to determine 2 sets of possible values for the variables.

new equation:

• Write an algebraic expression that is always greater than 3n 1 2 if n is a whole number. Substitute 3 values into the expression 3n 1 2 and into your expression to see if it seems right.

expression: _______________________

substituteto replace the value of a variable with a specific amount


Leaps and Bounds Using Variables, pathway 3


GuiDED

Mathematicians use letters as variables to represent unknown values. Steig is modelling an algebraic equation that uses a variable.

• You can use a variable to represent a single unknown value.

Recall equations such as 4 3 ❚ 5 12, which means,“If you multiply a number by 4, you get 12.”

You can use a letter variable instead of ❚ to represent the unknown number: 4n 5 12 means the same thing as 4 3 ❚ 5 12.

When you solve the equation 4n 5 12, n 5 3.

• You can also use variables to create an equation that represents a relationship. In a relationship, there is more than one set of values that make the equation true.

Suppose a rectangle is 10 cm long, but you don’t know its perimeter or width. You could represent the relationship using the equation P 5 10 1 w 1 10 1 w.

You could also use P 5 20 1 2w, because the expression 10 1 w 1 10 1 w is equivalent to 20 1 2w.

There are multiple possible values for the variables that make a relationship true.

For example, when you solve P 5 20 1 2w for w 5 2, P 5 24. If w 5 5, then P 5 30.

• Each equation above has an equals sign that means the algebraic expressions on both sides of the equation have the same value.

In 4 1 n 5 12, 4 1 n has the same value as 12.

In P 5 20 1 2w, P has the same value as 20 1 2w.

Using Variables Pathway 3

You will need• algebra tiles: x-tiles

and 1-tiles

• You can use any letter to represent a variable without changing the meaning of the expression or equation.


• When you solve an equation, you determine the value of the variable that makes the equation true. e.g.,

If 3n 5 6, then n 5 2. If j 5 2w and w 5 1,

then j 5 2. If w 5 2,then j 5 4, and so on.

Remember

algebraic equationa statement that uses an equals sign to show that 2 quantities have the same value

algebraic expressiona phrase that involves a set of numbers, operation signs, and variables




• You can model algebraic expressions using algebra tiles.

Each rectangle tile represents the variable, and each square tile represents 1. Light grey tiles show positive amounts, and dark grey tiles show negative amounts. The tile model below shows 2w 1 20.

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2w 20

The model shows why you cannot just combine 2w and 20 to determine the sum of 2w 1 20, because 2w and 20 are represented by different-sized tiles.

• You can use algebraic expressions to solve problems.

To calculate the perimeter of a 10 cm long rectangle, you can substitute a value for the width (w ) into the expression 20 1 2w.

For a width of 2 cm (w 5 2), the perimeter is 10 1 2 1 10 1 2 5 24 cm.

For w 5 5, it is 10 1 5 1 10 1 5 5 30.

• How do you know that the equation w 1 3 5 w 1 2 1 1 represents a relationship?

Try These 1. Are the variables used to represent an unknown value in the equation

or are they used in an equation that to describes a relationship? Circle the correct answer.

a) 30 4 k 5 6 unknown value relationship

b) 4 3 3 5 m unknown value relationship

c) j 1 10 5 5 1 j 1 5 unknown value relationship

d) m 5 10 2 n unknown value relationship

e) 5 5 10 2 r unknown value relationship

2. Match each expression to a meaning. One expression has no match.

half of a number

the difference between 20 and half of a number

20 more than double a number

2 more than 20 times a number

the difference between 20 and double a number

2n 1 20

20 2 2n

20 4 2 2 2n

n 4 2

20 2 n 4 2

20n 1 2

substituteto replace the value of a variable with a specific amount




3. Write an algebraic expression for each word expression.

a) 10 more than a number ________________

b) 10 less than 3 times a number ________________

c) double a number divided by 3 ________________

d) the area of a rectangle with a length of 30 cm and a width of w

________________

4. Model each expression using algebra tiles. Sketch the tiles.

a) k 1 3 c) 3 1 2k

b) 3k 1 2 d) 2k 2 3

5. Determine the value of each expression.

a) For w 5 8, 4w 1 2 5 ________.

b) For s 5 4, 6 2 s 5 ________.

c) For s 5 9, 3s 1 7 5 ________.

d) For m 5 2, 12 2 m 5 ________.

• The area of a rectangle is length 3 width.

Remember




6. Write an expression equivalent to each by making it simpler.You might use algebra tiles to help you.

a) 3 1 m 1 4 1 m 5 ________________

b) 8 1 h 1 2h 1 10 5 ________________

c) 7 2 w 2 w 1 9 5 ________________

d) 2 2 k 1 4 1 2k 5 ________________

7. Write an equation to represent each situation.

a) The number of legs is double the number of people.

________________________

b) If you divide a number by 8, you get 5.

________________________

c) If you double a number and add 8, you get the same amount as tripling the number and adding 4. ________________________

d) To multiply a number by 4, you can double its double.

________________________

e) If you triple a number and add 4, you get 22.

________________________

f) The total cost of some stamps is the price of one stamp, 54¢, multiplied by the number of stamps.

________________________

8. a) For Question 7 parts b), c), and e), solve the equations to determine the value of the variable.

__________________ __________________ __________________

b) For Question 7 parts a), d), and f), solve the equation to determine a possible value for the variables.

__________________ __________________ __________________

9. If you had to explain to someone the difference between an algebraic expression and an equation, what would you say?

Much of the math that you will do in secondary school is based on working with algebraic equations and expressions.

FYI




solving problems using equations 1 - wordpress.com · 500 20 25 it will take 25 weeks to save $550....

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