Creating
Definitions
Definitions
Perpendicular lines are two different lines that intersect at a right, or 90 degrees, angle.
Give two examples and two non-examples of perpendicular lines.
Answers will vary.
The first three terms in a sequence are 17, 26, 35.
What is the fifth term of the sequence?
the fifth term is 53
Look at the two sentences below:
"If a triangle is a right triangle, then the triangle has an acute angle."
"If a triangle has an acute angle, then the triangle is a right triangle."
Which of these two sentences, if any, are true? Why?
The first statement is true. There is no counterexample to the first statement because every right triangle has three angles where one angle measures 90 degrees and the other two angles measure less than 90 degrees (or acute). The second statement is false. A counter-example would be: “An equilateral triangle has an acute angle, but the triangle is not a right triangle.”
Katie and Tim baked identical pizzas. Katie cut her pizza into 8 equal slices and Tim cut his pizza into 6 equal slices. They each ate two (2) slices of their pizza.
Can we conclude that they ate the same amount of pizza? How come?
No, 2/8 doesn't equal 2/6. Their pizzas were cut differently, although the pizza itself is identical.
Use a diagram to prove that (x + 3)(x + 2) = x2 + 5x + 6. (see slides if needed)
The area of the larger rectangle is equal to the sum of the areas of the four smaller rectangles. So, (x + 3)(x + 2) = x2 + 5x + 6.
What single digit appears most frequently between and including the numbers 1 and 1,000?
The most common digit is 1! Every number 1–9 appears the same number of times in every ten numbers. But because we included the number 1,000, there’s an extra occurrence of the number 1. In total, the number 1 appears 301 times, and every other number appears 300 times.
Use the two-part structure to define the term "parallel lines."
Parallel lines are two different lines that extend forever in both directions and never intersect.
The first three terms in a sequence are 14, 10, and 6.
Write a rule to find the number in the nth term of the pattern.
-4n + 18
Take a look at the sentence below:
"If a figure is a rectangle, then the figure has 2 pairs of parallel sides."
What are the condition and the conclusion of this statement?
Condition: "If a figure is a rectangle."
Conclusion: "then the figure has 2 pairs of parallel sides."
Jack and Nikos are eating pizza. If they’re still hungry, they’ll go to Chipotle. If they go to Chipotle, Jack will order a burrito bowl. The other day, Jack and Nikos went to Chipotle.
Can we conclude that they had pizza before going to Chipotle?
nope! It is not guaranteed.
Use the diagrams to prove that (4/5)(1/2) = (4/10).
(See Ms. B's slides).
Once the shapes overlap, we have 10 rectangles total. Only 4 of them are in the double-shaded region. Thus, we have 4/10 rectangles total.
When I add six to eleven, I get five. Why is this correct?
Think about the numbers on a clock. When it is 11 a.m., adding six hours makes it 5 p.m.
Explain the difference between a square and a rectangle (use the two-part structure!).
Both a rectangle and a square have 4 sides and four 90-degree angles. However, a square must have 4 sides of equal length while a rectangle only requires opposite sides to be of equal length.
The first three terms in a pattern are 13, 78, and 468.
What is the fifth term in the pattern?
the fifth term is 16,848.
Take a look at the sentence below:
"If two rectangles have the same length and width, then the area of the rectangles is equal."
What is the converse of this statement?
Converse: "if the area of two rectangles is equal, then the rectangles have the same length and width."
A manufacturer claims that the average weight of an Oreo cookie is 11.3 grams. Mihailo purchased 5 different packages of Oreos and weighed them. The weights of an individual Oreo cookie ranged from 11.1 grams to 11.5 grams.
Did Mihailo prove or disprove the manufacturer’s claim?
Neither! You cannot prove a claim with examples.
What fraction of the figure is shaded? Show your work.
(see Ms. B's slides)
12/36 or 1/3
Using only addition, add eights (8's) to get the number 1,000.
888 + 88 + 8 + 8 + 8 = 1,000
Explain the difference between perimeter and circumference (use the two-part structure!).
Perimeter refers to the distance around the outside of a closed shape. When that shape is a circle, the perimeter is called the circumference.
The first three terms in a pattern are 5, 8, and 13.
What is the 18th term in the pattern? (see slides for visual)
the 18th term is 328.
(182 + 4 = 328)
Take a look at the statement below:
"If a triangle is equilateral, then the triangle doesn't have a right angle."
Is this a biconditional statement? Explain.
No. The converse of the statement is "if a triangle doesn't have a right angle, then the triangle is equilateral." The triangle could still be isosceles or scalene. It's not guaranteed to be equilateral.
Keen claims "the third side of a triangle must be greater than 7 in. if the sides are 3 and 4 inches."
Is this statement true or false? If true, prove the statement. If false, provide a counterexample.
False! The third side of a triangle needs to be less than 8 inches since the sides are 3 and 5 inches (3 + 5 = 8).
Six people shake hands with each other exactly one time. How many handshakes were there in total? Use the diagram to support your answer.
(See Ms. B's slides)
15 handshakes
"Two years ago, I was three times as old as my brother was. In three more years, I will be twice as old as my brother."
How old are the brothers now?
They are 17 and 7 years old.
Two years ago, they were 15 and 5. In three years, they will be 20 and 10.
Explain the difference between area and surface area (use the two-part structure!).
Area refers to the square units inside a 2D shape whereas surface area refers to the square units needed to wrap a 3D shape.
The first three terms of a sequence are 12, 20, and 30.
Write a rule to find the nth term of the pattern. (see slides for visual)
(n+2)(n+3) or n2 + 5n + 6
Take a look at the statement below:
"If a number is divisible by 10, then the last digit of a number is 0."
Is this a biconditional statement? Explain.
Yes, this is a biconditional statement. For the converse ("if the last digit is 0, then the number is divisible by 10"), it applies to other numbers but the number is still divisible by 10. So, the converse holds true.
Ms. B claims "multiples of 5 always alternate between odd and even numbers."
Is this statement true or false? If true, prove the statement. If false, provide a counterexample.
It is true. Ms. B will check the proof (students can't have a specific example as their proof).
Use the diagram to show or explain that a2 - b2 is equal to (a + b)(a - b).
(see Ms. B's slides)
Ms. B will check your answer.
A grandma left half her money to her granddaughter and half of that amount to her grandson. She left a sixth to her brother, and the remainder, $1,000, to the dogs’ home. How much did she leave altogether?
Adding 1/2, 1/4, and 1/6 tells us that the total is a fraction of twelfths (2 + 4 + 6 = 12). You can also think about it as 6/12, 3/12, 2/12, which equals 11/12. If the remainder is $1,000, that must be 1/12, so the total is $12,000.
Some student absences are excused if they are due to “extenuating circumstances.” How might you define “extenuating circumstances” in a way that is fair to students, their families, and the school staff?
Extenuating circumstances are extreme circumstances outside of one’s control that could be related to medical conditions, natural disasters, or other traumatic events.
The first three terms in a sequence are 2, 6, and 12.
Write a rule to find the nth term of the pattern. (see slides for visual)
n2 + n
Take a look at the statement below:
"If a quadrilateral is a rectangle, then the diagonals are congruent and bisect each other."
Is this a biconditional statement? Explain.
At a local restaurant, 402 customers were served. 299 customers ordered french fries and 136 customers ordered tater tots. If 70 customers ordered both french fries and tater tots, how many customers didn't have a side dish?
37 customers had no side dish.
Prove that the area of a trapezoid is ½ (h)(b1+b2). Hint: Use parallelograms to help you!
(See Ms. B's slides).
If you rotate a second trapezoid and stick it on the side of the first diagram, we get that the base is b1+b2 and that the height is still h. So, our p-gram has an area of h(b1+b2). Because our original trapezoid is half that amount, we can divide our formula by 2.
A small number of cards have been lost from a complete pack. If I deal among four people, three cards remain. If I deal among three people, two remain and if I deal among five people, two cards remain. How many cards are there?
There are 47 cards.
(there are 52 cards in a pack. Find the multiples of four, three, and five closest to that number, and see if any of them are the same.)