Find the area of each square. Each grid square represents 1 square unit.
Find the length of a side of a square if its area is:
Find the area of a square if its side length is:
Evaluate $(3.1 \times 10^4) \boldcdot (2 \times 10^6)$. Choose the correct answer:
Noah reads the problem, “Evaluate each expression, giving the answer in scientific notation.” The first problem part is: $5.4 \times 10^5 + 2.3 \times 10^4$. Noah says, “I can rewrite $5.4 \times 10^5$ as $54 \times 10^4$. Now I can add the numbers: $54 \times 10^4 + 2.3 \times 10^4 = 57.3 \times 10^4$.” Do you agree with Noah’s solution to the problem? Explain your reasoning.
Select all the expressions that are equivalent to $3^8$.
A square has an area of 81 square feet. Select all the expressions that equal the side length of this square, in feet.
$\frac{81}{2}$
$\sqrt{81}$
9
$\sqrt{9}$
3
Write the exact value of the side length, in units, of a square whose area in square units is:
Square A is smaller than Square B. Square B is smaller than Square C.
The three squares’ side lengths are $\sqrt{26}$, 4.2, and $\sqrt{11}$.
Find the area of a square if its side length is:
Here is a table showing the areas of the seven largest countries.
| country | area (in km2) | |
|---|---|---|
| row 1 | Russia | $1.71 \times 10^7$ |
| row 2 | Canada | $9.98 \times 10^6$ |
| row 3 | China | $9.60 \times 10^6$ |
| row 4 | United States | $9.53 \times 10^6$ |
| row 5 | Brazil | $8.52 \times 10^6$ |
| row 6 | Australia | $6.79 \times 10^6$ |
| row 7 | India | $3.29 \times 10^6$ |
Select all the expressions that are equivalent to $10^{\text-6}$.
Decide whether each number in this list is rational or irrational.
$$\frac {\text{-}13}{3}, \text{ }0.1234, \text{ }\sqrt{37}, \text{ -} 77, \text{ -} \sqrt{100}, \text{ -} \sqrt{12}$$
Which value is an exact solution of the equation $m^2=14$?
7
$\sqrt{14}$
3.74
$\sqrt{3.74}$
A square has vertices $(0,0), (5,2), (3,7)$, and $(\text-2,5)$. Which of these statements is true?
The square’s side length is 5.
The square’s side length is between 5 and 6.
The square’s side length is between 6 and 7.
The square’s side length is 7.
Rewrite each expression in an equivalent form that uses a single exponent.
The graph represents the area of arctic sea ice in square kilometers as a function of the day of the year in 2016.
The high school is hosting an event for seniors but will also allow some juniors to attend. The principal approved the event for 200 students and decided the number of juniors should be 25% of the number of seniors. How many juniors will be allowed to attend? If you get stuck, try writing two equations that each represent the number of juniors and seniors at the event.
Plot each number on the $x$-axis: $\sqrt{16},\text{ } \sqrt{35},\text{ } \sqrt{66}$. Consider using the grid to help.
Use the fact that $\sqrt{7}$ is a solution to the equation $x^2 = 7$ to find a decimal approximation of $\sqrt{7}$ whose square is between 6.9 and 7.1.
Graphite is made up of layers of graphene. Each layer of graphene is about 200 picometers, or $200 \times 10^{\text-12}$ meters, thick. How many layers of graphene are there in a 1.6-mm-thick piece of graphite? Express your answer in scientific notation.
Here is a scatter plot that shows the number of assists and points for a group of hockey players. The model, represented by $y = 1.5 x + 1.2$, is graphed with the scatter plot.
The points $(12, 23)$ and $(14, 45)$ lie on a line. What is the slope of the line?
Explain how you know that $\sqrt{37}$ is a little more than 6.
Explain how you know that $\sqrt{95}$ is a little less than 10.
Explain how you know that $\sqrt{30}$ is between 5 and 6.
Plot each number on the number line: $$6, \sqrt{83}, \sqrt{40}, \sqrt{64}, 7.5$$
Mark and label the positions of two square root values between 7 and 8 on the number line.
Select all the irrational numbers in the list. $$\frac23, \frac {\text{-}123}{45}, \sqrt{14}, \sqrt{64}, \sqrt\frac91, \text-\sqrt{99}, \text-\sqrt{100}$$
Each grid square represents 1 square unit. What is the exact side length of the shaded square?
For each pair of numbers, which of the two numbers is larger? Estimate how many times larger.
The scatter plot shows the heights (in inches) and three-point percentages for different basketball players last season.
Here is a diagram of an acute triangle and three squares.
Priya says the area of the large unmarked square is 26 square units because $9+17=26$. Do you agree? Explain your reasoning.
$m$, $p$, and $z$ represent the lengths of the three sides of this right triangle.
Select all the equations that represent the relationship between $m$, $p$, and $z$.
The lengths of the three sides are given for several right triangles. For each, write an equation that expresses the relationship between the lengths of the three sides.
Order the following expressions from least to greatest.
$25\div 10$
$250,\!000 \div 1,\!000$
$2.5 \div 1,\!000$
$0.025\div 1$
Which is the best explanation for why $\text-\sqrt{10}$ is irrational?
$\text- \sqrt{10}$ is irrational because it is not rational.
$\text- \sqrt{10}$ is irrational because it is less than zero.
$\text- \sqrt{10}$ is irrational because it is not a whole number.
$\text- \sqrt{10}$ is irrational because if I put $\text- \sqrt{10}$ into a calculator, I get -3.16227766, which does not make a repeating pattern.
A teacher tells her students she is just over 1 and $\frac{1}{2}$ billion seconds old.
Use the areas of the two identical squares to explain why $5^2+12^2=13^2$ without doing any calculations.
Each number is between which two consecutive integers?
$\sqrt{10}$
$\sqrt{54}$
$\sqrt{18}$
$\sqrt{99}$
$\sqrt{41}$
Give an example of a rational number, and explain how you know it is rational.
Write each expression as a single power of 10.
Andre is ordering ribbon to make decorations for a school event. He needs a total of exactly 50.25 meters of blue and green ribbon. Andre needs 50% more blue ribbon than green ribbon for the basic design, plus an extra 6.5 meters of blue ribbon for accents. How much of each color of ribbon does Andre need to order?
Find the exact value of each variable that represents a side length in a right triangle.
A right triangle has side lengths of $a$, $b$, and $c$ units. The longest side has a length of $c$ units. Complete each equation to show three relations among $a$, $b$, and $c$.
$c^2=$
$a^2=$
$b^2=$
What is the exact length of each line segment? Explain or show your reasoning. (Each grid square represents 1 square unit.)
In 2015, there were roughly $1 \times 10^6$ high school football players and $2 \times 10^3$ professional football players in the United States. About how many times more high school football players are there? Explain how you know.
Evaluate:
Here is a scatter plot of weight vs. age for different Dobermans. The model, represented by $y = 2.45x + 1.22$, is graphed with the scatter plot. Here, $x$ represents age in weeks, and $y$ represents weight in pounds.
Which of these triangles are definitely right triangles? Explain how you know. (Note that not all triangles are drawn to scale.)
A right triangle has a hypotenuse of 15 cm. What are possible lengths for the two legs of the triangle? Explain your reasoning.
In each part, $a$ and $b$ represent the length of a leg of a right triangle, and $c$ represents the length of its hypotenuse. Find the missing length, given the other two lengths.
For which triangle does the Pythagorean Theorem express the relationship between the lengths of its three sides?
Andre makes a trip to Mexico. He exchanges some dollars for pesos at a rate of 20 pesos per dollar. While in Mexico, he spends 9000 pesos. When he returns, he exchanges his pesos for dollars (still at 20 pesos per dollar). He gets back $\frac{1}{10}$ the amount he started with. Find how many dollars Andre exchanged for pesos and explain your reasoning. If you get stuck, try writing an equation representing Andre’s trip using a variable for the number of dollars he exchanged.
A man is trying to zombie-proof his house. He wants to cut a length of wood that will brace a door against a wall. The wall is 4 feet away from the door, and he wants the brace to rest 2 feet up the door. About how long should he cut the brace?
Select all the sets that are the three side lengths of right triangles.
8, 7, 15
4, 10, $\sqrt{84}$
$\sqrt{8}$, 11, $\sqrt{129}$
$\sqrt{1}$, 2, $\sqrt{3}$
For each pair of numbers, which of the two numbers is larger? How many times larger?
A line contains the point $(3,5)$. If the line has negative slope, which of these points could also be on the line?
Noah and Han are preparing for a jump rope contest. Noah can jump 40 times in 0.5 minutes. Han can jump $y$ times in $x$ minutes, where $y = 78x$. If they both jump for 2 minutes, who jumps more times? How many more?
The right triangles are drawn in the coordinate plane, and the coordinates of their vertices are labeled. For each right triangle, label each leg with its length.
Find the distance between each pair of points. If you get stuck, try plotting the points on graph paper.
Which line has a slope of 0.625, and which line has a slope of 1.6? Explain why the slopes of these lines are 0.625 and 1.6.
Write an equation for the graph. 
Write an equivalent expression that doesn’t use a cube root symbol.
Find the distance between each pair of points. If you get stuck, try plotting the points on graph paper.
Here is a 15-by-8 rectangle divided into triangles. Is the shaded triangle a right triangle? Explain or show your reasoning.
Here is an equilateral triangle. The length of each side is 2 units. A height is drawn. In an equilateral triangle, the height divides the opposite side into two pieces of equal length.
Find the positive solution to each equation. If the solution is irrational, write the solution using square root or cube root notation.
$t^3=216$
$a^2=15$
$m^3=8$
$c^3=343$
$f^3=181$
For each cube root, find the two whole numbers that it lies between.
Find the value of each variable, to the nearest tenth.
A standard city block in Manhattan is a rectangle measuring 80 m by 270 m. A resident wants to get from one corner of a block to the opposite corner of a block that contains a park. She wonders about the difference between cutting across the diagonal through the park compared to going around the park, along the streets. How much shorter would her walk be going through the park? Round your answer to the nearest meter.
Andre and Jada are discussing how to write $\frac{17}{20}$ as a decimal.
Andre says he can use long division to divide $17$ by $20$ to get the decimal.
Jada says she can write an equivalent fraction with a denominator of $100$ by multiplying by $\frac{5}{5}$, then writing the number of hundredths as a decimal.
Do both of these strategies work?
Which strategy do you prefer? Explain your reasoning.
Write $\frac{17}{20}$ as a decimal. Explain or show your reasoning.
Write each fraction as a decimal.
$\sqrt{\frac{9}{100}}$
$\frac{99}{100}$
$\sqrt{\frac{9}{16}}$
Write each decimal as a fraction.
$\sqrt{0.81}$
0.0276
$\sqrt{0.04}$
10.01
Find the positive solution to each equation. If the solution is irrational, write the solution using square root or cube root notation.
$x^2=90$
$p^3=90$
$z^2=1$
$y^3=1$
$w^2=36$
$h^3=64$
Here is a right square pyramid.
What is the measurement of the slant height $\ell$ of the triangular face of the pyramid? If you get stuck, use a cross section of the pyramid.
What is the surface area of the pyramid?
Elena and Han are discussing how to write the repeating decimal $x = 0.13\overline{7}$ as a fraction. Han says that $0.13\overline{7}$ equals $\frac{13764}{99900}$. “I calculated $1000x = 137.77\overline{7}$ because the decimal begins repeating after 3 digits. Then I subtracted to get $999x = 137.64$. Then I multiplied by $100$ to get rid of the decimal: $99900x = 13764$. And finally I divided to get $x = \frac{13764}{99900}$.” Elena says that $0.13\overline{7}$ equals $\frac{124}{900}$. “I calculated $10x = 1.37\overline{7}$ because one digit repeats. Then I subtracted to get $9x = 1.24$. Then I did what Han did to get $900x = 124$ and $x = \frac{124}{900}$.”
Do you agree with either of them? Explain your reasoning.
How are the numbers $0.444$ and $0.\overline{4}$ the same? How are they different?
$\frac{2}{3}$
$\frac{126}{37}$
Write each decimal as a fraction.
$0.\overline{75}$
Write each fraction as a decimal.
$\frac{5}{9}$
$\frac{5}{4}$
$\frac{48}{99}$
$\frac{5}{99}$
$\frac{7}{100}$
$\frac{53}{90}$
Write each decimal as a fraction.
$0.\overline{7}$
$0.\overline{2}$
$0.1\overline{3}$
$0.\overline{14}$
$0.\overline{03}$
$0.6\overline{38}$
$0.52\overline{4}$
$0.1\overline{5}$
$2.2^2 = 4.84$ and $2.3^2 = 5.29$. This gives some information about $\sqrt 5$.
Without directly calculating the square root, plot $\sqrt{5}$ on all three number lines using successive approximation.
