7th Grade MathJanuary 4 – January 8

(Mon/Tues) What are we doing today?

• Unit 4 Lesson 4

• Note: This is a new book. It should be the last book you picked up at the start of the year.

• Unit 3 Exam is not yet in the gradebook. I intend to update the gradebook later this week. We will go over the exam at a later time.

• MATHia is updated in the gradebook. It will be due again at the end of this week.

• Semester 1 ends on January 29th. So, we have 4 weeks before the end of the semester. As we get closer to the deadline, I will discuss makeup options.

• …trying to take over the world

Learning Targets

• I can use the distributive property to rewrite an expression like

𝑥 +1

2𝑥 as 1 +

1

2𝑥.

• I understand that “half as much again” and “multiply by 3

2” mean the

same thing.

Activity 4.2 (Part 1 & 2)

Activity 4.2 (Part 1 & 2 – ANSWER)

• In each row, we can divide the initial distance by 2 to find the additional amount walked. To find the total, we need to add this amount to the initial distance.

• For example, for the first row the initial distance is 10. So, we have 10

2= 5 for the additional distance

walked. Then, we can add 10 + 5 = 15 to find the total distance.

• In each row, we can follow this process to complete the table.

Activity 4.2 (Part 3)

• We still need to know what goes in the final box. Do you agree with Mai or Kiran? Why?

Activity 4.2 (Part 3 – ANSWER)

• Both students are correct as the expressions 𝑥 +

1

2𝑥 and

3

2𝑥 are the

same thing written differently.

• Let’s show this by using the first row of our table (𝑥 = 10):

𝑥 +1

2𝑥 → 10 +

1

210 = 10 + 5 = 15

3

2𝑥 →

3

210 = 15

• If you try this with another row, it will also lead to identical solutions.

Why is 𝑥 +1

2𝑥 the same as

3

2𝑥?

Why is 𝑥 +1

2𝑥 the same as

3

2𝑥?

• When we have a variable, such as x, it is the same as writing 1x (or 1 times x).

• Notice how x is the same as 1

2𝑥 +

1

2𝑥 as the

two halves would add to become 1.

• If we have 𝑥 +1

2𝑥, it is the same as having

3

2𝑥

or 1.5𝑥 as we have 1

2𝑥 three times (as shown

in the blue boxes).

• Notice how we can add the numbers in front of the variables to simplify an expression. In general, we usually want expressions to be written as simply as possible. However, it is important to be able to use the expression written either way.

Activity 4.3 (Part 1)

Activity 4.3 (ANSWER)

• For Part A, the answer is Diagram C. Notice how the y value must be smaller here as Mai ate less. So, we can eliminate A and B. In Diagram C, the y

value is 1

3less.

• For Part B, the answer is Diagram B. Notice how the y value must be larger here as Han biked more.

So, it must be A or B. In Diagram B, the y value is 2

3more.

• For Part C, the answer is (again) Diagram C. Again, the y value must be smaller. In Diagram C, the y

value is 2

3of the size of the x value.

• To help illustrate these relationships, I have labeled Diagrams B and C. In Diagram C, I just assumed each box was 1 unit. So, the x value is 3. The y value is 2. Using those numbers, would these values work for Part A and Part C?

Summary

Practice Problems (#1 & #2)

Practice Problem #1 (ANSWER)

• For Part 1, the answer is Diagram B as the y value is

1

4smaller.

• For Part 2, the answer is Diagram A as the y value is

3

4larger. It is also the only one where the y value is larger.

• For Part 3, the answer is Diagram C as the y value is

1

4of

the size of the x value.

Practice Problem #2 (ANSWER)

• The correct answers are C, D, and F.

• If Elina was 12 miles and then

walked 1

4of that distance further,

then she walked a total of 15 miles.

• In each choice, you can calculate the value of the expression and select which are equal to 15.

• Note: If needed, you could use Desmos to help calculate these values.

(Wed/Thurs) What are we doing today?

• Unit 4 Lesson 5

• Note: This is a new book. It should be the last book you picked up at the start of the year.

• Unit 3 Exam is not yet in the gradebook. I intend to update the gradebook later this week.

• If you have not completed Assignment #5, do so ASAP.

• Semester 1 ends on January 29th. So, we have 4 weeks before the end of the semester. As we get closer to the deadline, I will discuss makeup options.

• …trying to take over the world

Learning Targets

• I can use the distributive property to rewrite an equation like 𝑥 + 0.5𝑥 = 1.5𝑥.

• I can write fractions as decimals.

• I understand that “half as much again” and “multiply by 1.5” mean the same thing.

Activity 5.2

Activity 5.2 (ANSWER)

• The answers, in order, are 0.36, 0.3ത6, and 0. 36.

• On the right, I have shown two of the problems solved using long division. The order of the division is important. You must divide the numerator (top) by the denominator (bottom).

• Since the 2nd and 3rd decimals repeat, we can use a bar to show which decimals places repeat. Notice in my answers how the bar is over the decimals places which repeat. For the 2nd problem, this is only the 6. For the 3rd problem, it is both the 3 and the 6 as both repeat.

Activity 5.2 (ANSWER – DESMOS)

• Fractions can also be converted in Desmos.

• Simply type the fraction using the / key (near the right shift button). For example, 9/25 is my first fraction.

• Hit Enter and you will get the decimal value for that fraction. Desmos will round off if the decimals does not terminate.

• This allows us to easily see that the fractions from greatest to least are 11

30,4

11, and

9

25.

Definition

Activity 5.3 (PART 1 ONLY)

Activity 5.3 (Diagram A – ANSWER)

• For Diagram A, the answers are “A decrease by ¼” and 𝑦 = 0.75𝑥.

Activity 5.3 (Diagram B – ANSWER)

• For Diagram B, the answers are “An increase by ¼” and 𝑦 = 1.25𝑥.

Equation Notes

• When working with equations, it is very common to use either decimals or improper fractions. In time, you should be able to determine if two equations, or expressions, or identical.

• For example, 𝑦 = 1.25𝑥 and 𝑦 =5

4𝑥 are the same equation written in two

different ways. This is because 1.25 and 5

4are equal. Of course, it is also true that

5

4= 1

1

4. However, equations are rarely written with mixed numbers.

• If the decimal or fraction before the x value is larger than 1, the y value is larger than the x value. The opposite is also true. For example, the equation 𝑦 = 1.25𝑥

means that y is 0.25 or 1

4larger than x because we know 1 +

1

4= 1.25.

Summary

Practice Problems (1a and 2)

Practice Problem #1 (ANSWER)

• The answers for each diagram are marked with their respective colors here.

• For the descriptions, notice how you start with x and then see how y changes. For example, Diagram A has x being 5/5 boxes. Then y is 3/5 boxes. That is a decrease of 2/5.

• For the equations, you need to recall that you must start with 1 and then add or subtract the fraction. For example, Diagram A is 1 – 2/5 which is 3/5 or 0.6. So that is where the decimal coefficient comes from.

Practice Problem #2 (ANSWER)

• The answer is 0.7 ∙ 80 which is circled.

• Notice “the family ate 0.3” means that the peanut butter decreased. If we start with 1 and subtract 0.3, we get 0.7 which is what is left.

• Choice B would tell us how much was eaten. Choice C is trying to trick you into thinking that less simply means to subtract. However, “0.3 of” indicates multiplication. Choice D would be what happens if we added 0.3 more peanut butter.

(Fri) What are we doing today?

• Review Unit 3 Exam

• Unit 3 Exam is not yet in the gradebook. However, they are graded and can be reviewed online now.

• MATHia is due by the end of Saturday.

• Semester 1 ends on January 29th. So, we have 4 weeks before the end of the semester. As we get closer to the deadline, I will discuss makeup options.

• …trying to take over the world