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| 10 Theory slides |
| 8 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Tadeo and his little brother Dylan decide to go out together. They get in the elevator and Dylan randomly presses buttons on the panel until Tadeo stops him.
The elevator went 3 floors up and then 7 floors down before arriving at the ground floor, or Floor 1. Which floor were Tadeo and his brother on when they got in the elevator?
not equalsign ≠. Consider the result of substituting x=1 into the equation. Some equations are impossible to solve. Consider the following example. x=x+1 Notice that it is impossible to get a number itself after adding 1 to that number. This means that this equation has no solution. Conversely, it is possible for an equation to have more than one solution. An equation can also have infinitely many solutions, which means that all values satisfy the equation. x+x=2x This equation will be true for all values of x since the expressions on the left-hand side and right-hand side are equivalent expressions.
Sometimes the solution set of an equation can be obtained from a replacement set, which is a list of possible solutions. The values from the replacement set are substituted into the equation and then checked to confirm whether they satisfy the equation or not. Consider the following equation and its replacement set. Equation: & x-5=3 Replacement Set: & {5,6,7,8} In the following table, each value is substituted into the equation and the equation is evaluated.
Value of x | Substitute | Are both sides equal? |
---|---|---|
5 | 5-5 ? = 3 | 0 ≠ 3 * |
6 | 6-5 ? = 3 | 1 ≠ 3 * |
7 | 7-5 ? = 3 | 2 ≠ 3 * |
8 | 8-5 ? = 3 | 3 = 3 ✓ |
As shown in the table, the only solution of the equation is 3.
Tadeo and his brother Dylan are very hungry and decide to go to one of the restaurants at the city center. There are four different kinds of combos on the menu.
Menu | Menu Price | Substitution into 2x+8=22 | Check |
---|---|---|---|
Combo I | 7 | 2( 7)+8? =22 | 22 = 22 ✓ |
Combo II | 9 | 2( 9)+8? =22 | 26 ≠ 22 * |
Combo III | 5 | 2( 5)+8? =22 | 18 ≠ 22 * |
Combo IV | 6 | 2( 6)+8? =22 | 20 ≠ 22 * |
As shown in the table, the other prices do not satisfy the equation. This means that Tadeo and his brother both ordered Combo I.
x= 7
Calculate power
Multiply
Subtract terms
Menu | Menu Price | Substitution into x^2-15x=-54 | Check |
---|---|---|---|
Combo I | 7 | ( 7)^2-15( 7)? =-54 49-105? =-54 | -56 ≠ -54 * |
Combo II | 9 | ( 9)^2-15( 9)? =-54 81-135 ? =-54 | -54 = -54 ✓ |
Combo III | 5 | ( 5)^2-15( 5)? =-54 25-75 ? =-54 | -50 ≠ -54 * |
Combo IV | 6 | ( 6)^2-15( 6)? =-54 36-90 ? =-54 | -54 = -54 ✓ |
According to the table, the equation has two solutions, x=9 and x=6. This means that one of the brothers ordered Combo II and the other one ordered Combo IV. Bon appetit!
After having lunch, Tadeo and his brother Dylan walk around the city center. They start to discuss their physical characteristics.
Differencemeans subtraction.
Tadeo's height is 12cm more than Dylan's height.
The word is
represents an equals sign, which means that the expressions to the left and right sides of this word are equal to each other. Tadeo is 168 centimeters tall, so write this number on the left-hand side of the equation in place of Tadeo's height.
Next, more than indicates the addition of numbers. Let x be the variable that represents Dylan's height to write an expression for the right-hand side of the equation.
There may be other ways to write this equation. Saying that Dylan's height is 12 centimeters less than Tadeo's height has the same meaning. Here, less than refers to subtraction. Consider how to rewrite the equation as a subtraction expression. x=168-12 Both of the equations can be used to find Dylan's height.
Dylan's weight can be found by using the translated equation. x=3/4 * 52 Note that there are other possible ways to write the fraction 34. 3/4=75/100=0.75 This means that the equation x=0.75 * 52 also represents the given sentence.
differencemeans subtraction. Let x be the Dylan's age.
This situation can also be represented by other equations. If Tadeo is 3 years older than Dylan, we know that his age is Dylan's age plus 3. This means that the equation can be written using addition as well. x+3=16 Both equations can be used to find Dylan's age. Also, keep in mind that there is not only one way to represent a situation by an equation. Equations can be rewritten as several different equations by rearranging the terms of the equation.
Dylan and Tadeo like playing collectible card games and video games. They stop at a game store and see what is new.
<listcircle icon="b">Tadeo started with $15 more than the 2 he has left after buying the games.
Dylan and Tadeo have lots of old toy cars from their childhood. They decide to sell their old cars in a secondhand shop. They hope that other children will play with their old cars!
Consider how to split $ 54 in groups of $ 3. This will require division — in other words, divide the total amount that will be earned by the price of a single car to find the number of cars sold. Let y be the number of cars. 54/3=y Now calculate the quotient to find the value of y. 54/3=18 ⇒ y=18 This means that they need to sell 18 toy cars to get $ 54. Keep in mind that the scenarios from Part A and Part B can be solved by using other methods. These solutions are just examples of one way of how to find the required information.
At the beginning of the lesson, Tadeo and his brother got into an elevator to go out together. Dylan randomly pressed the elevator buttons until Tadeo stopped him.
The ground floor is represented as Floor 1. The boys went 7 floors down before getting off the elevator.
After the boys got in the elevator, it first went up 3 floors, then down 7 floors. Start by drawing a diagram to represent this situation.
Keep in mind that the ground floor is represented as Floor 1. Since the boys went 7 floors down before getting off the elevator, they must be at the 1+7=8^\text{th} floor after going 3 floors up.
Write the given sentence as an equation.
When writing words as equations, we look for keywords such as is
or equals
to find the placement of the equals sign.
5 less than a number b equals 9.
↓
=
Another keyword in our sentence is less than,
which indicates subtraction.
5 less than a number b equals 9.
⇓ ↓
- =
Now let's rewrite the other terms 5, the number b, and 9 to complete our equation.
The equation b-5=9 represents the given sentence.
Let's start by examining the given sentence again. Note that the word sum
means addition.
The sum of a number y and 3 is 13.
⇓
+
Next, we place the equals sign where the keyword is
stands.
The sum of a number y and 6 is 13.
⇓ ⇓
+ =
Finally write the terms y, 6, and 13 to complete the equation.
The equation y+6=13 represents the given word sentence.
We will decide which option does not represent the given equation. Let's start by examining the given options.
Sentences |
---|
7 subtracted from a number is 31. |
A number decreased by 7 is 31. |
A number minus 7 is the same as 31. |
A number is 7 less than 31. |
31 equals the difference between a number and 7. |
Let's write each sentence as an equation and check whether it matches the given equation. When writing sentences as equations, we look for keywords such as is
or equals
to find the placement of the equals sign. There are also some other keywords to decide the place of the operational signs. In this case, we want to use subtraction, so let's look for keywords that indicate subtraction.
difference
less than
subtracted from
decreased by
Since all of the options use the variable x, we will also use x to indicate a number
. With this in mind, let's write one of the options as an equation.
We can write the other sentences as equations by following the same steps.
Sentences | Equation |
---|---|
7 subtracted from a number is 31. | x - 7 = 31 |
A number decreased by 7 is 31. | x - 7 = 31 |
A number minus 7 is the same as 31. | x - 7 = 31 |
A number is 7 less than 31. | x = 31 - 7 |
31 equals the difference between a number and 7. | 31 = x - 7 |
As we can see from the table, the sentence A number is 7 less than 31.
needs to be represented with a different equation.
x=31-7
is not the same as
x-7=31
A group of 6 friends are planning to have a fun day together. They will choose one out of three different activities. The equations in the table represent the number of people x who can participate in each activity for $18.
Activity | Cost ($ ) |
---|---|
Balloon Ride | 3x-42=18 |
Amusement Park | 5x-22=18 |
Zoo Trip | 2x+6=18 |
We are asked to determine which activity the friends should choose if they want to spend exactly $18 and all do the same activity. There are 6 people in the friends group, so let's substitute x=6 into each equation and see which of them produce true statements. Let's start with the equation for the balloon ride.
Notice that x=6 is not a solution to this equation, so the 6 friends cannot join a balloon ride for $18. Let's apply the same thought process and substitute x=6 into the equations for the amusement park and zoo trip.
Activity | Cost ($) | Substitute x=6 | Multiply | Add or Subtract | Is x=6 a Solution? |
---|---|---|---|---|---|
Balloon Ride | 3x-42=18 | 3( 6)-42? =18 | 18-42? =18 | -24=18 * | No |
Amusement Park | 5x-22=18 | 5( 6)-22? =18 | 30-22? =18 | 8 ≠ 18 * | No |
Zoo Trip | 2x+6=18 | 2( 6)+6? =18 | 12+6? =18 | 18=18 ✓ | Yes |
We found that x=6 is a solution to the equation that represents the cost of the zoo trip. This means that if the friends want to spend exactly $18 and all do something together, they should choose the zoo trip.
A scale balances with four violet blocks on one side and one orange x-block on the other side. This situation is represented by the following equation. 4=x Heichi places 16 more violet blocks to the left side. The scale is now unbalanced.
We know that the following scale is balanced when four violet blocks are on the left side and one orange x-block on the right side.
We want to find a number that balances the scale again when multiplied by x.
Let's keep adding 4 violet blocks to the left side and 1 orange x-block to the other side until we get 20 violet blocks. This method will keep the scale balanced!
As we can see, if there are 20 violet blocks on the left side of the scale, we need 5 orange x-blocks to keep the scale balanced. This means that we need to multiply the x-blocks by 5. Now we can complete the given equation. 20=5* x