Free math worksheets

Evaluate. a) \(-14-19\) b) \(22-(-11)\) c) \(-35-(-15)\) d) \(8-23\)

A submarine is at \(-150\,\text{m}\) relative to sea level and then descends another \(75\,\text{m}\). Write an integer addition equation and find its new position.

Evaluate each expression and simplify the result. a) \(45-72\) b) \(-18+(-12)\) c) \(\frac{5}{6}-\frac{1}{6}\) d) \(-\frac{3}{8}-\frac{1}{8}\)

A weather station records a temperature of \(-9\,^\circ\text{F}\) one winter morning. During the morning, the temperature rises by \(14\,^\circ\text{F}\). What temperature does the station now show?

Mr. Weber's checking account has a balance of \(-\$180\). He deposits \(\$250\) in cash. What is the new account balance?

A type of plastic loses its flexibility and becomes brittle at \(-35\,^\circ\text{C}\). A new material mixture raises this threshold by \(12\) degrees. At what temperature does the new plastic become brittle?

A hiker begins a climb in a valley where the temperature is \(4\,^\circ\text{F}\). By the time the hiker reaches a mountain shelter, the temperature has dropped a total of \(15\) degrees. What is the temperature at the shelter?

What number is \(58\) greater than \(-33\)? Write and evaluate an equation.

Evaluate the expression. Show the value of the quantity in parentheses before finding the final value. \(7 \frac{5}{12} - \left(3 \frac{5}{6} + 2 \frac{1}{2}\right)\)

Which number is exactly halfway between \(-2.4\) and \(3.6\)? a) \(0.6\) b) \(0.8\) c) \(1.0\) d) \(1.2\)

Evaluate each expression. Pay attention to signs and grouping symbols. a) \(-18+42-(-15)\) b) \(125-(250-75)\) c) \(-15-[-40+(12-22)]\)

Calculate the value of the expression and write the result in simplest form: \(\frac{5}{12} - \left(\frac{1}{4} + \frac{1}{3}\right)\)

Calculate: \(\left(\frac{3}{8} - \frac{5}{6}\right) + \frac{11}{12}\)

Calculate each expression. Write each answer as a fraction in simplest form or as a mixed number. a) \(3 \frac{5}{12}+1 \frac{1}{4}\) b) \(6 \frac{2}{5}-4 \frac{7}{10}\) c) \(2 \frac{3}{4}+0.125\)

Calculate each expression. Write each answer as a mixed number in simplest form. a) \(3 \frac{2}{5}+4 \frac{5}{6}\) b) \(8 \frac{1}{4}-5 \frac{5}{8}\) c) \(10-\left(2 \frac{1}{3}+4 \frac{3}{5}\right)\)

Rearrange and group the terms to make the calculation efficient. Write the equivalent regrouped expression you use, then evaluate it. \(5\frac{3}{8} + 4\frac{2}{5} - 2\frac{3}{8} + 1\frac{3}{5}\)

For each expression, rewrite any fraction as a terminating decimal before evaluating. If an expression has no fraction, state that no conversion is needed. a) \(0.25 + \frac{1}{2}\) b) \(0.4 - \frac{3}{5}\) c) \(0.75 - 1\) d) \(1.1 + \frac{9}{10}\)

Lucas and Mia evaluate \(0.125 + \frac{3}{4} + 0.25 + \frac{1}{8}\) in different ways. Lucas converts the fractions to decimals. Mia converts the decimals to fractions and rearranges the addends. Carry out both methods. Which method seems more efficient here? Explain.

Evaluate \(\frac{5}{6} + 0.4 + \frac{1}{6} + 1.1\). Decide whether to convert every value to fractions or decimals, or to combine compatible terms separately. Briefly justify your choice.

Evaluate the expression by converting the fractions to terminating decimals and then showing one regrouping that creates convenient decimal pairs. \(\frac{2}{5} + 1.35 - \frac{3}{4} + 0.1\)

Evaluate the expression and write the result as a decimal: \(\frac{7}{10}-\left(\frac{2}{5}+\frac{1}{2}\right)\)

Find the error in the calculation and give the correct result. \(7\frac{1}{4} - 2.5 = (7 - 2) + (0.25 + 0.5) = 5 + 0.75 = 5.75\)

For each expression, choose a fraction or decimal form that avoids unnecessary work. Write the conversion or rewrite you chose before evaluating. a) \(\frac{3}{4} + 0.15\) b) \(0.8 - \frac{1}{5}\) c) \(-\frac{2}{3} + 0.5\) d) \(1.125 - \frac{1}{8}\)

Evaluate each expression. For every part, first write the exact fraction-to-decimal conversion you use; then calculate. a) \(0.45 + \frac{3}{20}\) b) \(\frac{5}{8} - 0.125\) c) \(-1.2 + \frac{1}{5}\)

Before evaluating, show an equivalent rearrangement and regrouping that creates two whole-number partial sums. Then evaluate and name the properties that justify changing the order and grouping. \(3.7 + \frac{2}{3} + 1.3 + 1\frac{1}{3}\)

Evaluate each expression. a) \(-4.8-2.35\) b) \(-\frac{5}{6}+\frac{1}{3}\) c) \(0.75+(-1.2)\) d) \(-\left(\frac{1}{4}-\frac{5}{8}\right)\)

Evaluate the expression: \((-4.2+1.7)-(0.8-2.3)\)

Calculate the value of the expression and write the result as a mixed number: \(\left(5 \frac{3}{4}-1 \frac{1}{2}\right)+2 \frac{5}{8}\)

A point starts at \(-2.3\) on a number line. It moves \(4.5\) units to the right and then \(1.4\) units to the left. Where does the point end?

Evaluate and write the result as a fraction in simplest form: \(\frac{2}{5}+\frac{1}{2}-\frac{3}{10}\)

Evaluate each sum or difference of rational numbers. a) \(-8.7+(-13.5)\) b) \(4.2-9.8\) c) \(-\frac{5}{9}+\frac{1}{6}\) d) \(\frac{2}{3}-\frac{3}{4}+\left(-\frac{1}{2}\right)\)

Evaluate each expression. Write the result as either a decimal or a fraction. a) \(\frac{3}{4}-1.25\) b) \(-0.6-\left(-\frac{1}{5}\right)\) c) \(2-\frac{1}{2}-0.75\) d) \(-\frac{1}{8}-0.125\)

The expressions are \(A=x-5.4\) and \(B=-3.2-y\). Evaluate them for \(x=2.1\) and \(y=-1.8\). Which value is greater?

During a winter night, the temperature drops to \(-12\,^\circ\text{C}\). By noon the next day, the temperature has risen to \(7\,^\circ\text{C}\). By how many degrees Celsius did the temperature increase?

Ms. Weber's bank account balance is \(-\$75.20\). She wants the balance to be exactly \(\$120.00\). How much money must she deposit?

Ms. Schneider's checking account balance is \(-\$125.50\). In the morning, she deposits \(\$350.00\). In the afternoon, she pays an \(\$80.00\) bill from the account. Find the end-of-day balance using one expression with rational numbers.

Evaluate \(T(a,b)=a-b\) for each pair of rational numbers. Complete the table using fractions in simplest form or mixed numbers. <table> <tr><td>\(a-b\)</td><td>\(-\frac{2}{3}\)</td><td>\(\frac{5}{6}\)</td></tr> <tr><td>\(-\frac{1}{2}\)</td><td></td><td></td></tr> <tr><td>\(\frac{3}{4}\)</td><td></td><td></td></tr> </table>

Determine whether each result is positive or negative without finding its exact value. For each part, give a sign rule or an absolute-value comparison that proves your choice; an exact numerical result alone is not a justification. a) \(-12+(-15)\) b) \(24-(-10)\) c) \(-0.5+0.3\) d) \(\frac{1}{2}-\frac{3}{4}\) e) \(-8-(-5)\)

Rewrite the expression as an equivalent sum that groups compatible fractions, then evaluate it. \(\frac{7}{15} - \frac{3}{4} + \frac{8}{15} - \frac{1}{4}\)

Evaluate without converting every number to the same form. Show an equivalent grouping that lets the decimals combine with decimals and the fractions combine with fractions. \(2.7 + \frac{2}{9} - 1.7 + \frac{16}{9}\)

For each expression, write a regrouped equivalent expression that creates convenient pairs, then evaluate. a) \(12.3 - 4.5 - 2.3 + 1.5\) b) \(\frac{3}{8} - 1.4 + \frac{5}{8} - 0.6\)

For each expression, rearrange and group terms to create at least one multiple of \(10\) or \(100\). Write the regrouped expression, then evaluate. a) \(34 + 128 - 134\) b) \(12.5 - 6.7 - 2.5 + 1.7\) c) \(-48 + 56 - 52 + 44\)

For each expression, show one regrouping that creates a whole-number partial sum or difference, then evaluate. a) \(15.6 - 7.8 - 2.2\) b) \(-4.9 + 12.5 - 5.1\) c) \(24.3 + 8.7 - 14.3 + 1.3\)

Use properties of addition to evaluate efficiently. \(7.25-4.8+\frac{11}{4}-1.2\)

Rearrange and group the terms to create convenient multiples of \(100\). Write the regrouped expression, then evaluate. \(-250 + 68 + 145 - 50 + 32 - 45\)

For each pair, write and evaluate an absolute-difference expression for the distance between the numbers. Then decide which pair is farther apart. Pair A: \(-12\) and \(8\) Pair B: \(-5.5\) and \(-20\)

For each expression, write one equivalent regrouping that produces convenient whole-number partial results, then evaluate. a) \(14.7 - 3.9 + 5.3 - 6.1\) b) \(-\frac{3}{8} + 12.5 - 5.5 + 0.375\) c) \(8.4 - (2.4 + 3.9) + 0.9\)

Evaluate the expression: \(\frac{2}{3}-\left(-\frac{1}{4}\right)+\frac{5}{6}-1\)

Evaluate each expression. Use common denominators where needed. a) \(\frac{4}{5}+\left(-\frac{1}{2}\right)\) b) \(-\frac{3}{4}-\frac{5}{6}\) c) \(-1.5+\frac{1}{4}\) d) \(\frac{7}{9}-\left(-\frac{2}{3}\right)\)

Complete each statement about how a subtraction difference changes. a) If the minuend increases by \(14\) and the subtrahend stays the same, the difference ______ by ______. b) If the minuend stays the same and the subtrahend decreases by \(9\), the difference ______ by ______. c) If both the minuend and the subtrahend increase by \(20\), then ______.

A subtraction expression is written as \(m - s = d\). Describe how \(d\) changes when both changes are made. a) Increase \(m\) by \(30\) and increase \(s\) by \(10\). b) Decrease \(m\) by \(15\) and increase \(s\) by \(15\). c) Increase \(m\) by \(12\) and decrease \(s\) by \(12\).

Start with \(300 - 120 = 180\). Decrease the minuend by \(50\) and decrease the subtrahend by \(30\). Use how these changes affect the difference to find the new difference. Then state how much it differs from the original difference.

A difference has a value of \(450\). The minuend is increased by \(120\), and the subtrahend is increased by \(50\). By how much does the difference change, and what is its new value?

Ms. Meyer has \(\$150\) in her checking account. A debit of \(\$215\) is then posted for a new desk. Explain why the new balance must be negative, and calculate the balance.

At \(6{:}00\) p.m. on a winter day, the temperature is \(4\,^\circ\text{F}\). By midnight, it drops \(7\,^\circ\text{F}\). During the early morning, it drops another \(2\,^\circ\text{F}\). What is the temperature in the morning?

For each sum, give a sign prediction and a brief rule or magnitude comparison that supports it. Then calculate the exact sum. a) \(-14+25\) b) \(18+(-30)\) c) \(-22+(-18)\) d) \(-45+45\)

Find the integer that makes each equation true. a) \(-12+\square=5\) b) \(\square+(-8)=-20\) c) \(15+\square=0\) d) \(\square+10=-3\)

Replace each box with \(<\), \(>\), or \(=\). For each part, give the sign rule or magnitude comparison that lets you decide without finding the exact sum. a) \(-35+40\mathbin{\square}0\) b) \(-12+(-15)\mathbin{\square}0\) c) \(25+(-25)\mathbin{\square}0\) d) \(-50+30\mathbin{\square}0\)

Find each sum. a) \(-38+92\) b) \(54+(-117)\) c) \(-256+(-144)\) d) \(-1025+375\)

Evaluate \(-120+450+(-380)\) from left to right. Report the intermediate sum after the first addition and the final sum.

Compare the values of Expressions A and B. Which is greater? Expression A: \(-1500+850\) Expression B: \(2100+(-2800)\)

A research submarine is \(450\,\text{m}\) below sea level. It rises \(185\,\text{m}\) to take a measurement. How far below sea level is the submarine after it rises?

Choose \(+\) or \(-\) for each box to make the equation true. a) \((\mathbin{\square}17)+13=30\) b) \(-24+(\mathbin{\square}16)=-40\) c) \(35+(\mathbin{\square}50)=-15\) d) \((\mathbin{\square}12)+(-28)=-16\)

Choose the missing sign in each equation. a) \(-42+12=\mathbin{\square}30\) b) \(15+(-45)=\mathbin{\square}30\) c) \((\mathbin{\square}60)+(-25)=35\) d) \(-11+(\mathbin{\square}11)=-22\)

For each sum, state the predicted sign and the rule or magnitude comparison that determines it. Then calculate the exact value. a) \(-415+625\) b) \(-874+(-126)\) c) \(2500+(-3750)\) d) \(-999+1001\)

Evaluate each sum from left to right. For each part, report the intermediate sum after the first addition and then the final sum. a) \(-120+250+(-80)\) b) \(500+(-750)+250\)

Evaluate both sides and replace each box with \(<\), \(>\), or \(=\). a) \(-45+20\mathbin{\square}-30+5\) b) \(100+(-150)\mathbin{\square}-80+20\) c) \(-12+(-13)\mathbin{\square}5+(-30)\)

Find the integer \(z\) that makes each equation true. a) \(50+z=-10\) b) \(z+(-120)=-200\) c) \(-15+25+z=0\)

A frozen-food warehouse has several storage zones. Zone A is kept at \(-18\,^\circ\text{C}\). Zone B is \(7\) degrees colder than Zone A. Zone C is \(12\) degrees warmer than Zone B. What are the temperatures in Zones B and C?

Insert \(+\) or \(-\) in each circle to make the equation true. a) \((\bigcirc 75)+(-25)=-100\) b) \((\bigcirc 120)+(+80)=-40\) c) \((+15)+(\bigcirc 50)=+65\) d) \((\bigcirc 9)+(+21)=+12\)

Find the missing addend in each equation. a) \(15+\square=8\) b) \(15+\square=-2\) c) \(15+\square=0\) d) \(15+\square=16\)

For each equation \(a+x=s\), find \(x\). a) \(-7+x=-15\) b) \(-7+x=4\) c) \(-7+x=-7\) d) \(-7+x=20\)

Find the missing addend. a) What number added to \(10\) gives \(-5\)? b) What number added to \(-20\) gives \(0\)? c) What number added to \(-4\) gives \(-12\)? d) What number added to \(0\) gives \(-8\)?

Evaluate \(154+(-289)+45\).

Evaluate \(-72+[148+(-205)]\).

Evaluate \(-120+55+(-80)+145\).

Which choice is closest to the value of \(-748+102+(-251)\)? A) \(-1100\) B) \(-900\) C) \(-400\) D) \(-650\)

Decide whether each statement is true or false. Justify with an example or counterexample. a) The sum of two negative numbers is always positive. b) Adding the opposite of \(0\) to any number leaves the number unchanged.

Find the missing addend. a) \(-520+\square=180\) b) \(75+\square=-25\)

Find \(x\). a) \(x+(-150)=-200\) b) \(x+300=50\)

Find \(x\). a) \(-12+x=-12\) b) \(44+x=0\)

For each pair, find \(s=a+b\), then decide whether \(s>a\). a) \(a=12\), \(b=-5\) b) \(a=-6\), \(b=-4\) c) \(a=-9\), \(b=10\)

On a winter afternoon, the temperature drops by \(9\,^\circ\text{F}\). That evening, the thermometer reads \(-5\,^\circ\text{F}\). What was the temperature before the drop?

Mr. Weber checks his banking app. A credit of \(\$125\) has been added, and his current balance is \(-\$240\). What was his balance immediately before the credit?

A remotely operated underwater vehicle descends \(150\,\text{m}\) from its initial position. It ends at \(-620\,\text{m}\), where sea level is \(0\,\text{m}\). What was the vehicle's position before it descended?

Rewrite each subtraction as addition of the opposite, and then evaluate. a) \(45-17\) b) \(-32-(-58)\) c) \(12-(-88)\) d) \(-74-26\)

Complete each equation by rewriting subtraction as addition of the opposite. a) \(-120-85=-120+(\square)=\square\) b) \(215-(-45)=215+(\square)=\square\) c) \(-63-(-163)=-63+(\square)=\square\) d) \(19-50=19+(\square)=\square\)

Rewrite each expression as a sum, evaluate it, and decide which value is greater. Expression A: \(-240-160\) Expression B: \(-240-(-160)\)

Order the expressions by value. Which expressions have equal values? 1. \(3-8\) 2. \(-2-3\) 3. \(12-4\) 4. \(-2-(-10)\) 5. \(0-5\) 6. \(15-7\)

Evaluate both sides and replace each box with \(<\), \(>\), or \(=\). a) \(-8-2\mathbin{\square}-8-(-2)\) b) \(5-5\mathbin{\square}-3-(-3)\) c) \(-12-4\mathbin{\square}-10-10\)

Evaluate both sides and replace each box with \(<\), \(>\), or \(=\). a) \(-58-22\mathbin{\square}-90+10\) b) \(72+(-84)\mathbin{\square}-5-(-7)\) c) \(-200+150\mathbin{\square}-10-45\)

Insert \(+\) or \(-\) in each box to make the equation true. a) \((\square14)+(-26)=-40\) b) \((-55)-(\square25)=-30\) c) \((+12)+(\square18)=-6\)

A research submarine is \(85\,\text{m}\) below sea level. It descends another \(40\,\text{m}\). Give its new position relative to sea level as an integer.

Evaluate each difference. First rewrite the subtraction as addition of the opposite. a) \((-410)-(+590)\) b) \((+1200)-(-800)\) c) \((-33)-(-33)\) d) \((-75)-(+25)\)

For each value of \(x\), evaluate \((-150)-x\). First rewrite the subtraction as addition. a) \(x=50\) b) \(x=-50\) c) \(x=150\) d) \(x=-200\)

Ms. Meyer's checking account balance was \(-\$45\) yesterday. Today, two more debits were posted: a \(\$30\) bill and a \(\$25\) shipping charge. What is her current balance?

Find each value. a) Find the sum of \(-42\) and \(-18\). b) Find the difference of \(-25\) and \(35\), with \(-25\) as the starting number.

Decide whether each statement is true or false. a) The sum of \(-15\) and \(15\) is \(0\). b) The difference of \(-40\) and \(-10\) is \(-50\). c) The sum of \(-9\) and \(-11\) is \(-20\).

How much less is \(-245\) than \(-112\)? Write an absolute-difference expression for the distance between the two values, evaluate it, and use that distance to answer the question.

A number plus \(-67\) equals \(25\). What is the number?

Evaluate \((-45)-(-18)+(-32)\).

Which subtraction expressions have a value of \(-24\)? Write all corresponding letters. A) \(16-40\) B) \(-12-12\) C) \(-30-(-6)\) D) \(0-(-24)\) E) \(-10-14\) F) \(-48-(-24)\)

First rewrite each expression without parentheses using simplified notation. Then evaluate. a) \((+56)+(-14)\) b) \((-33)-(-17)\) c) \((-120)+(-80)\) d) \((+200)-(+45)\)

Rewrite the expression as a sum. Then show an equivalent rearrangement or grouping that makes the calculation efficient before evaluating. \(-140+512-(-40)\)

Rewrite the expression as a sum. Then show an equivalent rearrangement or grouping that makes the calculation efficient before evaluating. \(385-119+615\)

Evaluate the expressions. Then group the expressions that have equal values. a) \(24-37\) b) \(37-24\) c) \(-(37-24)\) d) \(-24+37\) e) \(-24-(-37)\) f) \(-37+24\)

Rewrite without parentheses using simplified notation, and then evaluate: \((-432)-(-218)\).

Rewrite without parentheses using simplified notation, and then evaluate: \((+675)+(-280)-(+145)\).

Rewrite each expression as a sum by replacing subtraction with addition of the opposite. Then evaluate. a) \(-215-(-45)-30\) b) \(-128-72+100\)

Rewrite each expression using addition only, and then evaluate. a) \(440-600-(-160)\) b) \(-19-(-81)-100\)

Evaluate \(48+(52-315)\).

Lena has four number cards: \(16\), \(8\), \(-8\), and \(24\). She may use addition signs, subtraction signs, and parentheses to form expressions. Each card may be used at most once in an expression. Write two different expressions that each have a value of \(24\).

Which expressions have equal values? Group expressions with the same value, and give that value. A: \(45-(-5)\) B: \(45+5\) C: \(45+(-5)\) D: \(45-5\) E: \(50-0\) F: \(50-10\)

Evaluate each expression. a) \(-2500+100{,}500\) b) \(-340{,}000-12{,}500\) c) \(50{,}200+(-45{,}200)\) d) \(12{,}300-(-7700)\)

Consider the expression \((560 - 140) + (320 + 180)\). Without fully evaluating the new expression, determine how its value changes if each of the four numbers is increased by \(15\). Explain your reasoning from the structure of the expression.

Consider the expression \((1200 + 800) - (600 - 200)\). Determine how its value changes if each of the four numbers is increased by \(25\). Explain your reasoning from the structure of the expression.

At an Arctic weather station, the outdoor temperature is \(-46\,^\circ\text{C}\) early in the morning. By noon, it has risen to \(-19\,^\circ\text{C}\). By how many degrees Celsius did the temperature increase?

An unmanned research submarine is at \(-780\,\text{m}\) relative to sea level. It must rise to a maintenance station at \(-125\,\text{m}\). How many meters must the submarine rise?

Solve for \(x\). a) \(x+85=42\) b) \(-14+x=36\) c) \(x-55=-19\) d) \(-72+x=-110\)

Evaluate each expression. a) \(72-115+43\) b) \(-56+88-32\) c) \(105-(-45)-200\)

Evaluate \(-135-(62-140)\).

Evaluate \(-135+48+35-18\) by regrouping efficiently.

Simplify the operation signs and signs attached to numbers, and then evaluate: \(-12-(-18)+(-25)\).

Imagine moving on a number line. Subtracting a positive number means moving left. Write and evaluate the subtraction for each situation. a) Start at \(20\) and subtract \(55\). b) Start at \(-15\) and subtract \(30\).

Evaluate each expression. a) \((-75)+(-25)\) b) \((-75)-(-25)\) c) \(75+(-25)\) d) \(75-(-25)\)

Find the missing value in each addition equation. a) \(-18+30=\square\) b) \(\square+(-12)=-20\) c) \(-45+\square=-15\)

Rewrite subtraction as addition of the opposite. Then use the commutative and associative properties to show a grouping that creates \(100\), and evaluate. \(47 - 19 + 53\)

Use sea level as \(0\,\text{m}\). a) A submarine is \(250\,\text{m}\) below sea level. Write its position as an integer. b) A climber is on a peak \(1850\,\text{m}\) above sea level. Write the elevation as an integer. c) A diver is at \(-15\,\text{m}\) and descends another \(10\,\text{m}\). Write the new position as an integer. d) Find the vertical distance between a bird at \(+12\,\text{m}\) and a fish at \(-8\,\text{m}\).

A parking garage labels street level as Level \(0\). Levels below street level are negative, and levels above street level are positive. a) A car is parked on Level \(-3\). Describe its location. b) A visitor parks on Level \(-1\) and takes the elevator up \(4\) levels. On which level does the visitor exit? c) Later, the visitor rides directly from that level to Level \(-2\). Represent the elevator's change in level with a signed number.

A mountain weather station tracks temperature changes during the day. In the morning, the temperature rises by \(x\,^{\circ}\text{C}\). In the afternoon, it falls by \(y\,^{\circ}\text{C}\). a) Write an expression for the net temperature change. b) Find the net change on each day. 1) Day 1: \(x=5.4\), \(y=3.1\) 2) Day 2: \(x=2.8\), \(y=4.5\) c) Explain what the sign of each result means.

A class fund starts the month with some savings. During the month, it earns \(\$e\) from a bake sale and spends \(\$a\) on art supplies. a) Write an expression for the change in the fund balance. b) Find the change when \(e=54\) and \(a=38\). Explain the result. c) Find the change when \(e=25\) and \(a=42\). Compare the ending balance with the starting balance.

A reservoir helps regulate a river. Each hour, \(z\,\text{m}^3\) of water flows into the reservoir, and \(a\,\text{m}^3\) flows out through the gates. a) Write an expression for the change in water volume after one hour. b) What does a value of \(0\) mean in this context? c) What does a negative value mean? d) Find the hourly change when \(z=1250\) and \(a=1310\).

Tim is saving for a skateboard that costs \(\$85\). He has already saved \(\$y\). a) Write an expression for the signed difference “price minus savings.” b) Evaluate the expression for \(y=62\) and for \(y=90\). Explain what each result means.

A hiker is at an elevation of \(h\) feet. The destination lodge is at an elevation of \(4000\) feet. a) Write an expression for the signed elevation difference “destination elevation minus current elevation.” b) Evaluate the difference for \(h=3200\) and for \(h=4500\). Interpret each result as an uphill or downhill change.

A bank account has a current balance of \(\$x\). A bill of \(\$y\) is withdrawn. a) Write an expression for the new account balance. b) Find the balance when \(x=125\) and \(y=40\). c) Find the balance when \(x=50\) and \(y=85\). Explain what a negative result means for the account holder.

Evaluate by rearranging and grouping the addends. Show the reordered expression, the two grouped partial sums, and the final value. \(17.4 + (-5.9) + 2.6 + (-4.1)\) Which property allows you to change the order of the addends?

Find each sum of rational numbers. 1) \((-17)+(+25)+(-13)\) 2) \((+4.8)+(-6.3)+(-2.5)\) 3) \(\left(-\frac{3}{10}\right)+0.75\) 4) \(\left(-2 \frac{1}{4}\right)+\left(-1 \frac{5}{8}\right)\)

For each pair of values, find \(c\) so that \(a+b+c=0\). a) \(a=-8.5\), \(b=3\frac{1}{4}\) b) \(a=\frac{2}{3}\), \(b=-\frac{1}{6}\) c) \(a=15.7\), \(b=-20\)

A weather station records morning and noon temperatures. Calculate each temperature change as \(\text{noon temperature}-\text{morning temperature}\), and enter a positive number, a negative number, or zero in the table. <table> <thead> <tr> <th>Day</th> <th>Morning temperature</th> <th>Noon temperature</th> <th>Temperature change</th> </tr> </thead> <tbody> <tr> <td>Monday</td> <td>\(+2\,^\circ\text{C}\)</td> <td>\(-3\,^\circ\text{C}\)</td> <td></td> </tr> <tr> <td>Tuesday</td> <td>\(-5\,^\circ\text{C}\)</td> <td>\(+1\,^\circ\text{C}\)</td> <td></td> </tr> <tr> <td>Wednesday</td> <td>\(-4\,^\circ\text{C}\)</td> <td>\(-4\,^\circ\text{C}\)</td> <td></td> </tr> <tr> <td>Thursday</td> <td>\(0\,^\circ\text{C}\)</td> <td>\(-6\,^\circ\text{C}\)</td> <td></td> </tr> </tbody> </table>

Evaluate each expression. 1) \((-18)-(+12)-(-20)\) 2) \((-5.4)-(-2.6)-(+1.2)\) 3) \((+3.7)-(+8.7)-(-5)\)

For each sum, write the grouping that creates an additive inverse or a whole-number partial sum, then evaluate. 1) \((-34) + 57 + 34\) 2) \(12.4 + (-6.8) + (-3.2)\) 3) \(-2\frac{1}{5} + 8 - 1\frac{4}{5}\)

Calculate \(x-y\) for each pair. a) \(x=-2.4\), \(y=5.6\) b) \(x=\frac{3}{5}\), \(y=-\frac{7}{10}\) c) \(x=-1\frac{1}{2}\), \(y=-2\frac{1}{4}\) d) \(x=-0.75\), \(y=0.75\)

The number lines show integer jumps. Find each number marked with a red question mark.

The number lines show integer jumps. Find the number marked with a red question mark in each part. a) Find the ending value. b) Find the starting value. c) Find the missing intermediate value.

Find the number marked with a red question mark on each number line.

A school snack stand tracks its monthly finances. The bar chart shows the profit (positive values) or loss (negative values), in dollars, for the first six months of the year. a) Find the total profit or loss after the six months. b) Were there more months with a profit or more months with a loss?

Find each ending value on the number lines. Write an addition equation for each jump.

Find each missing value on the number lines.

Find the missing ending value on each number line.

Find the missing ending value for each negative jump.

Use the number line to find the missing ending value.

The number line shows the peak elevation and the vertical distance from the base to the peak of an underwater mountain. How deep is the base of the mountain below sea level?

The number-line diagram shows Max’s starting bank account balance and a deposit. What is his new balance?

The number line shows the change produced by subtracting \(-5\) from the displayed starting value. a) Find the missing ending value. b) Write the subtraction equation represented by the diagram. c) Explain why the move is to the right even though the operation is subtraction.

Point \(A\) is a starting position and point \(B\) is an ending position on the number line. a) Read \(A\) and \(B\) as fractions in simplest form. b) Write the signed change from \(A\) to \(B\) as \(B-A\), and evaluate it. c) Does the sign of the change match the direction of movement? Explain.

Find the fraction exactly halfway between \(\frac{2}{3}\) and \(\frac{4}{5}\). Write your answer in simplest form.

Compare the structure of expressions \(A\) and \(B\) before calculating. \(A=-15+(30-45)\) \(B=-15+30-45\) 1) Find the value of each expression. 2) What rule about removing parentheses is illustrated here? 3) How would \(B\) need to change if \(A\) were \(-15-(30-45)\) instead?

Point \(P\) is at \(-3.4\) on a number line, and point \(Q\) is at \(1.2\). a) Find the number \(m\) exactly halfway between \(P\) and \(Q\). b) Is \(m\) an integer? Is it a rational number? c) List all integers strictly between \(-3.4\) and \(1.2\).

Compare \(\frac{19}{8}\) and \(2.37\). a) Which number is greater? b) Find the decimal exactly halfway between the two numbers.

Find the number exactly halfway between each pair. a) Between \(\frac{5}{8}\) and \(0.63\). Give the answer as a decimal. b) Between \(0.7\) and \(0.71\). Give the answer as a fraction in simplest form.

Find the decimal exactly halfway between \(\frac{1}{5}\) and \(\frac{1}{4}\). Then give one decimal greater than \(\frac{1}{5}\) but less than your midpoint.

Consider \(\frac{3}{8}\), \(0.3\), \(\frac{2}{5}\), and \(0.38\). a) Convert the fractions to decimals and order all four numbers from least to greatest. b) Give one decimal strictly between the two greatest values from part a).

Calculate the expression and write the result as a fraction in simplest form: \(0.4 + \frac{1}{3} - 15\%\) What problem occurs if you try to calculate this expression exactly using only terminating decimals and no fractions?

Find the rational number \(x\) exactly halfway between \(-\frac{1}{2}\) and \(0.3\). Then determine whether \(x\) is greater than or less than \(-\frac{1}{8}\). Justify your comparison.

Calculate each expression. Write each answer as a fraction in simplest form or as a mixed number. a) \(1 \frac{3}{8} + \frac{7}{8} - \frac{5}{8}\) b) \(\frac{4}{15} - \left(\frac{7}{15} + \frac{12}{15}\right)\) c) \(2 \frac{1}{12} - \frac{5}{12} - \frac{11}{12}\)

Calculate the value of the nested expression: \(\frac{2}{5} - \left(\frac{1}{2} - \left(\frac{3}{10} + \frac{1}{4}\right)\right)\)

Compare the results of calculations \(A\) and \(B\). Insert \(<\), \(>\), or \(=\), and justify your answer with calculations. \(A=5 \frac{1}{3}-2 \frac{1}{2}\) \(B=1 \frac{3}{4}+1 \frac{1}{6}\)

Evaluate the expression using the correct order of operations. Write the answer as a mixed number. \(7 \frac{1}{5}-\left(2 \frac{1}{2}+1 \frac{3}{4}\right)+0.8\)

Simplify and remove the parentheses, then write an equivalent expression whose grouping makes the final calculation efficient. Evaluate the expression. \(\left(6\frac{4}{15} + \frac{18}{24}\right) - \left(1\frac{4}{15} - \frac{1}{4}\right)\)

Rewrite the subtraction as addition of opposites. Then show an equivalent rearranged and regrouped sum that makes the calculation efficient, evaluate it, and name the properties that justify changing the order and grouping. \(\left(15\frac{5}{6} + 3\frac{1}{4}\right) - \left(2\frac{5}{6} + 1\frac{1}{2} + 1\frac{3}{4}\right)\)

Evaluate each expression. Before calculating, state which representation you will use for each nonmatching pair and show the first rewritten expression. a) \(\left(\frac{1}{4} + 0.5\right) - 1\) b) \(2.75 - \left(1\frac{1}{2} + 0.25\right)\) c) \(\frac{1}{3} + 0.2 + \frac{1}{6}\)

Evaluate the expression. Briefly explain why fractions or decimals are the better representation. \(1\frac{1}{3} + 0.25 - \frac{7}{12}\)

Consider \(\frac{3}{4}+\frac{2}{3}-\frac{1}{2}\). a) Give a benchmark estimate that decides whether the result is greater than or less than \(1\). Show the benchmark values you used. b) Calculate the exact value as a fraction. c) State whether the exact result is consistent with your estimate.

Which expression has the greater value? Evaluate both expressions and justify your answer. Expression A: \(\left(\frac{3}{4}-1.25\right)\cdot(-4)\) Expression B: \(2.5\div\frac{1}{2}-6\)

Check the following work for an error: \(\frac{4}{5}-\left(\frac{1}{2}-\frac{1}{10}\right)=\frac{4}{5}-\frac{1}{2}-\frac{1}{10}=\frac{8}{10}-\frac{5}{10}-\frac{1}{10}=\frac{1}{5}\). Explain whether the work is correct. If it is not, identify the error and calculate the correct result.

Evaluate the expression: \(\left(\frac{2}{3}-1.5\right)+\left(-\frac{5}{6}+0.25\right)\)

Evaluate the expression: \(\left(-4.5\div\frac{9}{10}\right)\cdot\left(\frac{2}{3}-1.2\right)\)

Evaluate the expression: \(\frac{5}{8}-\left[(-2.5)\cdot\left(\frac{1}{5}-0.6\right)\right]\div\left(-\frac{1}{2}\right)\)

Two students evaluate \(4.8+2\frac{1}{4}+1.2+\frac{3}{4}\). - Jordan converts the fractions to decimals and adds from left to right. - Maya reorders and groups the addends as \((4.8+1.2)+(2\frac{1}{4}+\frac{3}{4})\). a) Evaluate the expression both ways. b) Explain which method is easier to do mentally.

Decide whether each statement is true or false. Justify your decision by evaluating both sides of the equation. a) \(-12.5+7.5=-(12.5-7.5)\) b) \(\frac{1}{2}-\left(\frac{1}{4}+\frac{1}{8}\right)=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}\)

Evaluate the expression. Write the result as a fraction in simplest form or as a mixed number: \(10-\left(3 \frac{1}{4}+2 \frac{5}{6}\right)-1 \frac{1}{12}\)

Evaluate the expression. Write the answer as a fraction in simplest form or as a mixed number: \(2 \frac{1}{4}-\frac{5}{6}-\frac{3}{8}\)

Find the missing value of \(a\), \(b\), or \(a+b\) in each part. a) \(a=-4.5\), \(b=2.1\), \(a+b=\square\) b) \(a=\square\), \(b=-7.8\), \(a+b=-3.2\) c) \(a=\frac{3}{4}\), \(b=\square\), \(a+b=-\frac{1}{8}\)

Complete the subtraction table and find \(x\) and \(y\). Each entry is the value at the left of its row minus the value at the top of its column. <table> <tr><td>\(-\)</td><td>\(1.2\)</td><td>\(y\)</td><td>\(-3.5\)</td></tr> <tr><td>\(4.5\)</td><td>\(3.3\)</td><td>\(7.2\)</td><td></td></tr> <tr><td>\(-2.8\)</td><td></td><td>\(-0.1\)</td><td>\(0.7\)</td></tr> <tr><td>\(x\)</td><td>\(-5.5\)</td><td></td><td>\(-0.8\)</td></tr> </table>

Insert \(<\), \(>\), or \(=\) to make each statement true. For each part, justify the comparison using a rational-number rule or magnitude comparison without giving the exact value of the expression. a) \(-4.5+4.5\;\dots\;0\) b) \(-123-(-122)\;\dots\;0\) c) \(-\frac{2}{3}+\left(-\frac{1}{3}\right)\;\dots\;0\) d) \(0.01-0.1\;\dots\;0\)

For each pair, decide which expression has the greater value. Justify the choice using signs, direction on a number line, or relative magnitude; do not give the exact final values. a) \((-15)+(-20)\) or \((-15)-(-20)\) b) \(0.4-0.6\) or \(-0.4+0.6\) c) \(-\frac{1}{8}-\frac{1}{4}\) or \(-\frac{1}{8}+\frac{1}{4}\)

Let \(x=-5\) and \(y=-2\). For each expression, first predict whether the result is positive or negative. Then find the exact value. a) \(xy\) b) \(x+y\) c) \(y-x\) d) \(\frac{x}{y}\)

Choose exactly three different numbers from the list whose sum is \(0\). Find two different solutions. \(0.7,\ -0.3,\ \frac{1}{5},\ -0.4,\ 0.1,\ -0.5\)

Choose exactly four numbers from the list. Place a plus or minus sign between consecutive chosen numbers so that the result is \(0.5\). \(1.2,\ -0.5,\ \frac{1}{4},\ -0.8,\ 0.1,\ -0.2\)

Two students evaluate \(-0.5 + \frac{3}{4} - 0.25 + \frac{1}{2}\) in different ways. Mia works from left to right. Lucas groups values that combine easily. a) Evaluate the expression using Mia’s method. b) Show an efficient grouping Lucas could use. c) Compare the original expression with \(-0.5 + \left(\frac{3}{4} - 0.25\right) + \frac{1}{2}\). Do the parentheses change the value? Explain.

For each expression, write an equivalent grouping that combines compatible rational numbers before evaluating. a) \(-\frac{3}{7} + \frac{5}{8} - \frac{4}{7} + \frac{3}{8}\) b) \(1\frac{2}{5} - 3\frac{1}{4} + \frac{3}{5} + \frac{1}{4}\) c) \(\frac{7}{10} - 0.45 + \frac{3}{10} - 0.55\)

Lina and Tim evaluate \(12.5 - 8.3 + 7.5 - 1.7\) in different ways. Lina groups the positive terms and the amounts being subtracted: \((12.5 + 7.5) - (8.3 + 1.7)\). Tim groups consecutive pairs: \((12.5 - 8.3) + (7.5 - 1.7)\). a) Evaluate the expression using both methods. b) Explain why Lina’s method is especially efficient here.

Evaluate each expression using properties of rational-number addition. For each part, show the reordered or regrouped expression that makes the calculation efficient. a) \(\frac{5}{9} - \frac{1}{3} + \frac{4}{9}\) b) \(\frac{3}{4} - 2.5 - \frac{7}{4} + 1.5\) c) \(-\frac{2}{7} + \frac{5}{6} + \frac{2}{7} - \frac{1}{6}\)

Rewrite each expression by removing parentheses or regrouping strategically, then evaluate. a) \(-3.25-(1.75-8)\) b) \(2\frac{1}{5}-\left(4.8+\frac{1}{5}\right)+0.8\) c) \(\left(\frac{5}{8}+1.4\right)-\left(\frac{1}{8}-0.6\right)\)

Use all four number cards \(-4\), \(-2\), \(6\), and \(8\), along with addition and subtraction signs, to write an expression with a value of \(0\). You may use parentheses. Then write a different expression using the same four cards that has a value of \(12\).

Insert an addition or subtraction sign in each blank to make the equation true. a) \(-7\;\square\;(-3)\;\square\;5=-9\) b) \(-7\;\square\;((-3)\;\square\;5)=1\)

A point is at \(-3.4\) on a number line. A second point at coordinate \(x\) is exactly \(7.2\) units away. a) Write an absolute-value equation that models the distance condition. b) Solve the equation by considering the two possible signed differences.

For each pair, write an absolute-difference expression and evaluate the distance. Give each answer as a decimal or a fraction in simplest form. a) \(-\frac{3}{4}\) and \(0.2\) b) \(1\frac{1}{2}\) and \(-\frac{2}{5}\)

A number line is drawn so that \(1\) unit is \(4\,\text{cm}\). a) How many centimeters from \(0\) should \(-1.25\) be placed, and in which direction? b) What number is exactly halfway between \(-\frac{1}{2}\) and \(\frac{3}{4}\)? Give the answer as a fraction in simplest form and as a decimal.

Evaluate each expression using properties of operations. For each part, show the cancellation or regrouping that makes the calculation short. a) \(17.4 - (5.9 + 17.4) + 0.9\) b) \(-\frac{2}{5} + (3.7 + 0.4) - 2.7\)

Rewrite and evaluate each expression. Pay attention to the mixture of fractions and decimals. a) \(\frac{1}{8}-0.25+0.875-1.75\) b) \(\left(\frac{5}{6}+1.2\right)-\left(\frac{1}{6}-0.8\right)\)

Place exactly one pair of parentheses in \(40-20-10-5\). The parentheses must contain at least two numbers. a) Find the value without parentheses. b) Place the parentheses so the value does not change. c) Place the parentheses so the value is \(25\).

Consider the expression \(50-25-10-5\). Determine whether placing exactly one pair of parentheses can make the value \(40\). The parentheses must enclose at least two numbers. Justify your answer by evaluating every possible placement.

Evaluate the expression using the order of operations: \(18\div(-3)-4\cdot(-2.5)\)

A weather station starts the day at \(45.5\,{}^\circ\text{F}\). The temperature then changes by \(-3.4\,{}^\circ\text{F}\), \(+8.8\,{}^\circ\text{F}\), \(-2.2\,{}^\circ\text{F}\), \(-5.7\,{}^\circ\text{F}\), and \(+6.1\,{}^\circ\text{F}\), in that order. a) What is the final temperature? b) What is the highest temperature reached during the day? c) What is the lowest temperature reached during the day?

A rain barrel holds \(50\) gallons. It starts the week with \(30.75\) gallons of water. During the week, the amount changes in this order: - Rain: \(+8.50\) gallons - Watering: \(-13.20\) gallons - Rain: \(+3.85\) gallons - Watering: \(-22.40\) gallons a) How much water is in the barrel at the end of the week? b) When the barrel reached its highest amount during the week, how many gallons of empty capacity remained?

A checking account starts with a balance of \(\$250.00\). These transactions occur in order: - Withdrawal: \(\$315.50\) - Deposit: \(\$120.00\) - Withdrawal: \(\$85.25\) a) What is the lowest balance reached during the sequence? b) What is the balance after the final transaction? c) How much must be deposited after the final transaction to bring the balance to exactly \(\$100.00\)?

Find the rational number that belongs in each blank. a) \(\frac{2}{3}+\square=-\frac{1}{6}\) b) \(\square-\frac{3}{5}=-1\) c) \(-\frac{1}{2}-\square=\frac{1}{8}\)

Write each expression and find its value. a) Decrease \(-18.5\) by the difference of \(12.4\) and \(15.9\). b) Add the opposite of \(7.2\) to the sum of \(-14.6\) and \(21.3\). c) Subtract three times \(4.5\) from the difference of \(-10\) and \(-25\).

A weather station in Colorado records these temperature changes from one reading to the next. At midnight, the temperature is \(27.5\,{}^\circ\text{F}\). <table> <tr><td>6:00 a.m.</td><td>\(+7.2\,{}^\circ\text{F}\)</td></tr> <tr><td>9:00 a.m.</td><td>\(+6.8\,{}^\circ\text{F}\)</td></tr> <tr><td>12:00 p.m.</td><td>\(+4.4\,{}^\circ\text{F}\)</td></tr> <tr><td>3:00 p.m.</td><td>\(-2.5\,{}^\circ\text{F}\)</td></tr> <tr><td>6:00 p.m.</td><td>\(-8.6\,{}^\circ\text{F}\)</td></tr> <tr><td>9:00 p.m.</td><td>\(-4.3\,{}^\circ\text{F}\)</td></tr> </table> a) Write one expression for the temperature at 9:00 p.m. and calculate it. b) Find the highest and lowest temperatures at the listed reading times, including midnight.

Lucas tracks his allowance account. At the beginning of April, the balance is \(\$28.50\). These transactions occur during the month: <table> <tr><td>Apr. 4</td><td>Movie and popcorn</td><td>\(-\$14.20\)</td></tr> <tr><td>Apr. 10</td><td>Gift from Grandma</td><td>\(+\$20.00\)</td></tr> <tr><td>Apr. 15</td><td>New video game</td><td>\(-\$35.00\)</td></tr> <tr><td>Apr. 22</td><td>Mowing a neighbor's lawn</td><td>\(+\$12.50\)</td></tr> <tr><td>Apr. 28</td><td>Pizza with friends</td><td>\(-\$6.80\)</td></tr> </table> a) Write an expression and find the account balance at the end of April. b) Determine whether the account balance was ever negative during April. If so, give the lowest balance.

A bulk-food store tracks changes in its oatmeal inventory in pounds. On Monday morning, the store has \(34.0\) pounds. During the week, these amounts are sold (negative) or delivered (positive): <table> <tr><td>Monday</td><td>\(-7.0\,\text{lb}\)</td></tr> <tr><td>Tuesday</td><td>\(-10.0\,\text{lb}\)</td></tr> <tr><td>Wednesday</td><td>\(+44.0\,\text{lb}\) (delivery)</td></tr> <tr><td>Thursday</td><td>\(-16.5\,\text{lb}\)</td></tr> <tr><td>Friday</td><td>\(-20.5\,\text{lb}\)</td></tr> </table> a) How many pounds of oatmeal remain on Friday evening? Write an expression. b) What was the greatest amount of oatmeal in the store at one time during the week?

In a subtraction expression, the minuend is increased by \(45\). How must the subtrahend change so that the difference is \(60\) greater than it was originally?

In a subtraction expression, the subtrahend is \(2458\). It is \(1035\) less than the minuend. a) Find the minuend. b) The minuend stays fixed. By how much must the subtrahend decrease for the difference to increase by \(150\)?

Three students evaluate \(1000 - 250 - 150 - 100\). - **Leo:** \(1000 - 250 = 750\), \(750 - 150 = 600\), \(600 - 100 = 500\). - **Mia:** \(250 + 150 + 100 = 500\), then \(1000 - 500 = 500\). - **Noah:** \(1000 - 250 = 750\), \(150 - 100 = 50\), then \(750 - 50 = 700\). Decide which methods are correct. Explain Noah’s error if his method is incorrect.

A magician says, “Start with my secret number. Add \(2450\), subtract \(1100\), and then add \(325\). The result is the least five-digit whole number.” Find the secret number.

A research submersible is at \(-120\,\text{m}\), where sea level is \(0\,\text{m}\). It descends another \(150\,\text{m}\) to take a measurement, then rises \(40\,\text{m}\). What is the submersible's final position relative to sea level?

Choose signs for \(20\) and \(45\) to make each equation true. Write the completed equation. a) \((\mathbin{\square}20)+(\mathbin{\square}45)=65\) b) \((\mathbin{\square}20)+(\mathbin{\square}45)=-25\) c) \((\mathbin{\square}20)+(\mathbin{\square}45)=25\) d) \((\mathbin{\square}20)+(\mathbin{\square}45)=-65\)

Insert either \(+\) or \(-\) in each box to make the equation true. a) \((-18)\mathbin{\square}(-7)=-11\) b) \((+25)\mathbin{\square}(-5)=+30\) c) \((-40)\mathbin{\square}(+10)=-50\) d) \((+12)\mathbin{\square}(-12)=0\)

Insert the missing signs so that each equation is true. a) \((\bigcirc100)+(\bigcirc40)+(-20)=-80\) b) \((+50)+(\bigcirc30)+(\bigcirc10)=+10\)

Estimate to decide which expression has the lesser value. Then calculate both values exactly. Expression 1: \(-3250+(-1780)\) Expression 2: \(-8410+3520\)

Estimate by rounding each addend to the nearest thousand, and then evaluate \(-4890+(-3120)+6950\) exactly.

Write an expression for each description and evaluate it. a) Find the difference of \(1500\) and \(850\), then subtract the opposite of \(150\). b) Add the absolute value of \(-75\) to the sum of \(-125\) and \(-200\).

Subtract the sum of \(-48\) and \(122\) from the difference of \(300\) and \(-50\). Write and evaluate the expression.

Match each sum to its exact result. Use estimation to find the pairs efficiently. Sums: a) \(-298+(-405)+(-102)\) b) \(-895+302+198\) c) \(512+(-251)+(-262)\) d) \(-1203+(-395)+(-102)\) Results: 1) \(-1\) 2) \(-1700\) 3) \(-805\) 4) \(-395\)

Decide whether each statement is true or false. Briefly explain. a) Start with any number, add \(12\), and then add the opposite of the original number. The result is always \(12\). b) The sum of a negative number and its opposite is always positive.

Maya claims, “The sum of two negative integers is always less than either addend.” Test the claim using \(-7+(-3)\), and then explain whether it is always true.

Let \(a=-8\) in \(a+b=s\). Give one integer value of \(b\) for each condition. a) \(s<a\) b) \(s=a\) c) \(s>a\)

Give one pair of integers with sum \(-32\) for each condition. a) Both addends are negative. b) The first addend is positive, and the second is negative. c) The first addend is negative, and the second is positive.

A submarine's net vertical change is \(-80\,\text{m}\), where positive changes mean upward movement and negative changes mean downward movement. Give one addition equation for each situation. a) The submarine descends during both stages. b) The submarine rises during the first stage and descends during the second stage.

Complete each equation so the sum is \(-50\). a) \(25+\square=-50\) b) \(-70+\square=-50\) c) \(\square+\square=-50\), using two identical addends.

Evaluate this claim: “Subtracting the absolute value of an integer from the integer always gives \(0\).” Decide whether the claim is true or false and justify your answer with examples.

Complete the calculation chain \(-120\xrightarrow{+45}\square\xrightarrow{-30}\square\xrightarrow{-(-60)}\square\).

Each equation is missing both a sign attached to a number and an operation sign. Insert \(+\) or \(-\) to make each equation true. Find all possibilities. a) \((\square8)\mathbin{\bigcirc}(-12)=+20\) b) \((-25)\mathbin{\bigcirc}(\square15)=-10\) c) \((\square40)\mathbin{\bigcirc}(+60)=-20\)

Complete the subtraction table. Each entry is the row value minus the column value. <table> <tr> <th colspan="2" rowspan="2"></th> <th colspan="4">Number being subtracted</th> </tr> <tr> <th>\(15\)</th> <th>\(-10\)</th> <th>\(-35\)</th> <th>\(60\)</th> </tr> <tr> <th rowspan="3">Starting number</th> <th>\(-20\)</th> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <th>\(45\)</th> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <th>\(-80\)</th> <td></td> <td></td> <td></td> <td></td> </tr> </table>

Evaluate \((-128)-[(+52)+(-97)]\).

Evaluate \([(-34)+(+86)]-[(-42)-(+58)]\) step by step.

Use the numbers \(-34\) and \(-16\). a) Find their sum. b) Subtract the sum from the opposite of \(50\). What is the result?

Use the numbers \(12\) and \(-18\). First find their sum and their difference, in that order. Then subtract the difference from the sum. What is the final result?

On a cold January day, temperatures are recorded at two locations in the mountains. At noon, the mountain station is \(-11\,^\circ\text{F}\), and at night it is \(-24\,^\circ\text{F}\). In the valley, the temperature is \(4\,^\circ\text{F}\) at noon and \(-6\,^\circ\text{F}\) at night. a) For each location, by how many degrees Fahrenheit did the temperature decrease from noon to night? b) Find the temperature difference between the mountain station and the valley at noon and at night.

A laboratory uses several cooling areas. The room temperature is \(22\,^\circ\text{C}\), a refrigerator is \(4\,^\circ\text{C}\), a freezer is \(-18\,^\circ\text{C}\), and a dry-ice chamber is \(-78\,^\circ\text{C}\). a) Write and evaluate an absolute-difference expression for the room-to-freezer temperature difference. b) Write and evaluate an absolute-difference expression for the refrigerator-to-dry-ice-chamber difference. c) Write absolute-difference expressions for the refrigerator-to-freezer and freezer-to-dry-ice-chamber differences. Which is greater?

Complete the account table. In each row, an amount is withdrawn from the previous balance to produce the new balance. <table> <tr><th>Previous balance</th><th>Withdrawal</th><th>New balance</th></tr> <tr><td>\(\$12\)</td><td>\(\$20\)</td><td>(1)</td></tr> <tr><td>\(-\$8\)</td><td>\(\$15\)</td><td>(2)</td></tr> <tr><td>(3)</td><td>\(\$25\)</td><td>\(-\$10\)</td></tr> </table>

Evaluate each difference, and then insert \(<\), \(>\), or \(=\) to make a true statement. a) \((-12)-8\mathbin{\square}-12\) b) \((-12)-(-8)\mathbin{\square}-12\) c) \(15-15\mathbin{\square}15\) d) \(15-(-15)\mathbin{\square}15\)

Consider the two expressions. Expression \(A\): \((-40)-25\) Expression \(B\): \((-40)-(-25)\) Without calculating exact values, explain which expression is greater. Use what you know about subtracting integers and compare each result with \(-40\).

Evaluate each expression with three numbers. First rewrite it without parentheses using simplified notation. a) \((-15)+(+25)-(-10)\) b) \((+40)-(+60)+(-20)\) c) \((-100)-(-30)-(+70)\)

Start at \(-50\) and perform these operations in order: 1. Subtract \(-30\). 2. Add \(-45\). 3. Subtract \(+15\). Write one expression with parentheses, simplify it, and find the final value.

Rewrite the expression as a sum. Then show an equivalent rearrangement and grouping that creates convenient partial sums before finding the final value. \(-2650+1433-350+567\)

Compare the values and insert \(<\), \(>\), or \(=\) in each box. a) \(18-30\mathbin{\square}-18+30\) b) \(-50-(-20)\mathbin{\square}-50+20\) c) \(45-(10+50)\mathbin{\square}45-10-50\) d) \(-(25-5)\mathbin{\square}-25-5\)

Three expressions have the same value. Which expression does not belong? Justify your choice by evaluating all four. A) \(-22+10-8\) B) \(-22+(8-10)\) C) \(10-(22+8)\) D) \(-(22-10)-8\)

Rewrite each expression as a sum, and then evaluate. a) \(25{,}000-60{,}000\) b) \(-10^4-5000\) c) \(8800-(-1200)\) d) \(-10^6-(-10^5)\)

Mr. Schmidt pays a \(\$545\) bill from his checking account. Soon afterward, a credit of \(\$210\) is deposited. His balance is then exactly \(-\$185\). What was his account balance before these two transactions?

An elevator in a parking garage first travels up \(5\) levels and then down \(8\) levels. It ends on Level \(-2\). On which level did the elevator start?

First estimate by rounding each number to the nearest ten. Then evaluate efficiently by rearranging and grouping terms. \(48+63-28+37-80\)

First make a reasonable estimate. Then evaluate efficiently by grouping positive and negative numbers. \(-135+420-65+580-300\)

Make a reasonable estimate, and then evaluate efficiently by rearranging or grouping terms. \(2750-438-1250+638-500\)

Rewrite without parentheses using simplified notation, and then evaluate: \(1000-(-450)+(-1250)-(+200)\).

Consider the expression \(-45+128-55+72\). a) Evaluate the expression efficiently by grouping terms. b) Replace the last number so that the expression has a value of \(0\). What should the new number be?

Consider the expression \(1250-3400+750-600\). a) Evaluate it efficiently by grouping terms. b) Replace the last term, \(-600\), so that the expression has a value of \(0\). Give the new signed number.

Evaluate each expression step by step. Follow the order of operations. a) \(26-[(41-55)+(82-14)]\) b) \([-18+(72-95)]-(31+29)\)

Evaluate each expression using the order of operations. a) \(-32-(74+26)+(-85+35)\) b) \((-94-126)-(-52+318)\)

Evaluate \(-240+115+(-60)+85\) efficiently. Name the properties used to regroup the addends.

Evaluate efficiently by rearranging and grouping terms: \(781-495-281+195-100\).

Evaluate \((12-45)-(150+5)\).

During one level of a video game, Tim's score changes several times. He first earns \(120\) bonus points, then loses \(250\) points, and finally earns another \(40\) points. His score at the end of the level is \(-30\). What was Tim's score at the beginning of the level?

At \(6{:}00\) a.m., the temperature is \(-4\,^\circ\text{F}\). It then rises \(7\,^\circ\text{F}\), falls \(2\,^\circ\text{F}\), rises \(3\,^\circ\text{F}\), falls \(8\,^\circ\text{F}\), and falls another \(5\,^\circ\text{F}\). Find the final temperature. Calculate efficiently.

A research submarine starts at \(-1250\,\text{m}\) relative to sea level. It rises \(450\,\text{m}\), descends \(200\,\text{m}\), rises \(800\,\text{m}\), and descends \(150\,\text{m}\). Find its new position. Calculate efficiently.

A club account starts at \(-\$240\). During the quarter, it receives \(\$1550\) in dues, pays \(\$600\) in rent, spends \(\$760\) on equipment, receives a \(\$450\) donation, and pays \(\$300\) for an event. Find the new balance. Calculate efficiently.

Let \(a=-45\), \(b=20\), and \(c=-30\). a) Write and evaluate an expression that subtracts \(c\) from the sum of \(a\) and \(b\). b) Write and evaluate an expression that subtracts the sum of \(a\) and \(b\) from \(c\). c) Compare the results. What general observation can you make?

A research team measures elevations relative to sea level. <table> <tr><td>Location</td><td>Elevation</td></tr> <tr><td>A (peak)</td><td>\(+340\,\text{m}\)</td></tr> <tr><td>B (meadow)</td><td>\(+115\,\text{m}\)</td></tr> <tr><td>C (village)</td><td>\(-25\,\text{m}\)</td></tr> <tr><td>D (lakeshore)</td><td>\(-142\,\text{m}\)</td></tr> <tr><td>E (cave floor)</td><td>\(-205\,\text{m}\)</td></tr> </table> For each part, write and evaluate an absolute-difference expression. a) Find the elevation difference between A and C. b) Find the vertical separation between D and E. c) Find the elevation difference between the highest and lowest locations.

Ms. Berger records her checking account balance at the end of each weekday. <table> <tr><td>Monday</td><td>\(+\$12\)</td></tr> <tr><td>Tuesday</td><td>\(-\$4\)</td></tr> <tr><td>Wednesday</td><td>\(-\$18\)</td></tr> <tr><td>Thursday</td><td>\(-\$9\)</td></tr> <tr><td>Friday</td><td>\(+\$22\)</td></tr> </table> a) Find the signed change from Monday to Tuesday and state whether the balance increased or decreased. b) Identify the lowest balance and write an absolute-difference expression for its distance from Friday's balance. c) Write the four absolute daily changes between consecutive days and identify the greatest one.

Which tasks have equal values? Sort the letters into the groups with values \(-25\) and \(-55\). (A) \(-40+15\) (B) \(-40-(-15)\) (C) Subtract \(15\) from \(-40\). (D) Add \(-15\) to \(-40\). (E) \(-40-15\) (F) Add \(15\) to \(-40\). (G) Subtract \(-15\) from \(-40\).

Find the sum \(1 + 2 + 3 + \dots + 79 + 80\) by pairing terms. 1) Name the two properties that let you reorder and regroup the addends. 2) How many pairs are formed when you pair the first term with the last term, the second term with the next-to-last term, and so on? 3) What is the sum of each pair? 4) Find the total sum.

An apple seller stacks apples in a triangular display. The top row has \(1\) apple, and each row below it has one more apple than the row above. The bottom row has \(40\) apples. 1) Write an expression for the total number of apples. 2) Find the total efficiently by pairing terms. 3) Explain why \(20 \cdot 41\) gives the total.

Find the sum of all even numbers from \(2\) through \(60\): \(2 + 4 + 6 + \dots + 58 + 60\) 1) How many terms are in the sum? 2) Pair terms that have the same sum. How many pairs are there, and what is the sum of each pair? 3) Find the total.

Evaluate each expression. a) \(-12{,}450+5670\) b) \(34{,}200-(-8950)\) c) \(-7300-12{,}800\) d) \(10^4-100{,}000\)

Insert \(<\), \(>\), or \(=\) to make each statement true. a) \(-3500+1200\mathbin{\square}-3500-1200\) b) \(-800-450\mathbin{\square}-800-(-450)\) c) \(15{,}000-25{,}000\mathbin{\square}-5000-6000\) d) \(-10^3+1\mathbin{\square}0\)

Write an expression and evaluate it. From the sum of \(-412\) and \(187\), subtract the difference of \(-256\) and \(-58\).

Leon claims, “When I subtract a negative number from another negative number, the result is always negative.” Disprove his claim with two examples: 1. One example with a positive result. 2. One example with a result of \(0\). Evaluate both differences.

Consider \((-14)-(-20)\). a) Evaluate the difference. b) Explain why the result is positive even though both numbers in the expression are negative.

Use the numbers \(-6\) and \(-10\). 1. Evaluate \((-6)-(-10)\). 2. Evaluate \((-10)-(-6)\). Compare the results. What do you notice about their signs?

Write an expression for the description and evaluate it. Subtract \(-150\) from the sum of \(-65\) and \(38\).

Write an expression for the description and evaluate it. Add the difference of \(16\) and \(-34\) to the difference of \(210\) and \(450\).

Rewrite the expression to make it easy to evaluate. Keep each operation sign with its number when you reorder the terms. \(864 + 257 - 364 + 43\)

Evaluate \(30-[(-12+25)-(8-15)]\).

Evaluate each expression using the order of operations. a) \(84-(112-250)\) b) \(-37+(-44+19)+60\) c) \(-(55-92)-(18+24)\)

Evaluate each expression with nested grouping symbols. a) \(15-[32-(14-50)]\) b) \(-21+[(-45+18)-(-12)]\)

Evaluate \(-35-(15-60)+(-20)\).

Evaluate \((120-250)+(85-(-15))\) step by step.

Rewrite the expression to make it easy to evaluate. Keep each operation sign with its number when reordering the terms. \(45 - 17 + 55 - 83 + 12\)

A checking account begins with a balance of \(\$120\). During the morning, \(\$150\) is withdrawn for a purchase, and then another \(\$40\) is withdrawn at an ATM. Use subtraction expressions to find the balance after each withdrawal.

Use the numbers \(-12,7,-3,0,-8,2\). a) Order the numbers from greatest to least using \(>\). b) Find the pair of numbers with the least sum. c) Find the two numbers that are farthest apart on a number line.

Use the numbers \(-10,4,-1,6,-5,3\). a) Which numbers lie strictly between \(-6\) and \(4\) on a number line? b) Order all six numbers from least to greatest using \(<\). c) Find a pair of numbers whose sum is \(5\).

A diving robot begins at \(-45\,\text{m}\) relative to sea level and descends \(15\,\text{m}\) each minute. After how many whole minutes will it first be below \(-110\,\text{m}\)?

Estimate by rounding to the nearest ten. Then evaluate \(128-57-43+72-110\) exactly using efficient regrouping.

Evaluate \(-42+(18-50)+60\).

Evaluate \(80-[-30-(25-45)]\).

Evaluate the four expressions, and then order their values from least to greatest. a) \(-110+(-40)\) b) \(-110-(-40)\) c) \(110+(-110)\) d) \(0-110\)

Rewrite the expression to make it easy to evaluate. Keep each subtraction sign with the number that follows it. \(154 + 78 - 54 - 28\)

Evaluate \(-123+57+265-77+43-65\) efficiently using properties of addition.

Add the difference of \(45\) and \(120\) to the sum of \(-33\) and \(-17\).

Subtract the sum of \(-215\) and \(90\) from the opposite of \(45\).

Find the sum of \(-15\), \(42\), and \(-10\). Subtract that sum from the difference of \(-100\) and \(50\).

Ms. Miller begins the week with \(\$85\) in her checking account. During the week, these transactions occur: 1. A \(\$120\) grocery purchase is debited. 2. A credit of \(\$50\) is deposited. 3. A \(\$30\) bill is paid. Calculate the balance after each transaction. What does the sign of the final balance mean?

A weather station records two temperature changes on each of three days. Find the final evening temperature for each day. 1. Monday begins at \(5\,^\circ\text{C}\). The temperature rises \(8\,^\circ\text{C}\), then falls \(10\,^\circ\text{C}\). 2. Tuesday begins at \(-3\,^\circ\text{C}\). The temperature rises \(6\,^\circ\text{C}\), then falls \(7\,^\circ\text{C}\). 3. Wednesday begins at \(-1\,^\circ\text{C}\). The temperature falls \(4\,^\circ\text{C}\), then rises \(5\,^\circ\text{C}\).

An elevator uses Level \(0\) for street level, positive numbers for levels above street level, and negative numbers for levels below street level. a) The elevator starts on Level \(2\), travels down \(5\) levels, and then travels up \(2\) levels. Where does it stop? b) The elevator is on Level \(-1\) and travels down \(3\) more levels. Where does it stop? c) The elevator is on Level \(-2\) and must travel to Level \(3\). How many levels must it travel upward?

For each movement on a number line, find the missing value and write the corresponding addition equation. a) Start at \(-5\) and move \(8\) units right. b) Start at \(3.5\) and move \(6\) units left. c) You end at \(-2\) after moving \(4\) units left from an unknown starting point. Find the starting point.

Analyze sums on a number line. a) Which sum lies farther left: \((-12)+7\) or \((-3)+(-3)\)? Calculate and compare. b) A point is at \(-4.8\). What number \(x\) must be added to move it to \(0\)? c) A point at \(-2.4\) is moved to its opposite by adding a number \(y\). Find \(y\) and write the complete equation.

Complete the table by finding each missing value of \(x\), \(y\), or \(x+y\). <table> <thead> <tr> <th>Item</th> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>5</th> </tr> </thead> <tbody> <tr> <td>\(x\)</td> <td>\(-7.2\)</td> <td>\(3 \frac{1}{4}\)</td> <td>\(-0.8\)</td> <td>\(-\frac{5}{6}\)</td> <td>\(-4.15\)</td> </tr> <tr> <td>\(y\)</td> <td>\(4.5\)</td> <td>\(-5 \frac{1}{2}\)</td> <td>\(\dots\)</td> <td>\(\frac{1}{3}\)</td> <td>\(4.15\)</td> </tr> <tr> <td>\(x+y\)</td> <td>\(\dots\)</td> <td>\(\dots\)</td> <td>\(-2\)</td> <td>\(\dots\)</td> <td>\(\dots\)</td> </tr> </tbody> </table>

Evaluate each expression, then order the results from least to greatest. a) \(-12.5+7.8\) b) \(-\frac{3}{5}-\frac{1}{2}\) c) \(4 \frac{1}{3}+\left(-2 \frac{5}{6}\right)\) d) \(-0.75-(-1.2)\)

Evaluate each sum using the commutative and associative properties. For every part, write the regrouped expression you use before giving the result. 1) \((-24) + 15 + (-16) + 35\) 2) \(6.7 + (-3.2) + 1.3 + (-1.8)\) 3) \(-\frac{3}{4} + 2\frac{1}{5} - 0.25 + 0.8\)

Evaluate each sum by choosing and showing pairings that avoid converting every term to one representation. 1) \(0.375 - 5\frac{1}{2} + \frac{5}{8} + 4.5\) 2) \(-8.1 + 2\frac{3}{7} + 1.1 + 5\frac{4}{7}\) 3) \(11\frac{1}{9} - 4.9 - 2\frac{1}{9} - 5.1\)

A small snack stand records its daily revenue and expenses. The daily net is \(\text{revenue}-\text{expenses}\). A positive result is a profit, and a negative result is a loss. a) Calculate the daily net for each day. b) Find the total net for all four days. <table> <thead> <tr> <th>Day</th> <th>Revenue</th> <th>Expenses</th> <th>Daily net</th> </tr> </thead> <tbody> <tr> <td>Monday</td> <td>\(\$145\)</td> <td>\(\$162\)</td> <td></td> </tr> <tr> <td>Tuesday</td> <td>\(\$210\)</td> <td>\(\$185\)</td> <td></td> </tr> <tr> <td>Wednesday</td> <td>\(\$95\)</td> <td>\(\$130\)</td> <td></td> </tr> <tr> <td>Thursday</td> <td>\(\$178\)</td> <td>\(\$178\)</td> <td></td> </tr> </tbody> </table>

At \(6{:}00\) a.m. on a winter day, the temperature is \(-6\,^\circ\text{F}\). a) By noon, the temperature rises \(11\,^\circ\text{F}\). What is the noon temperature? b) By late evening, the temperature drops \(8\,^\circ\text{F}\) from the noon value. What is the evening temperature? c) Find the difference between the highest and lowest of the morning, noon, and evening temperatures.

Julia and Tom compare their checking account balances. - Julia has \(\$12\), so her balance is \(+\$12\). - Tom's account is overdrawn by \(\$8\), so his balance is \(-\$8\). a) What is the difference between their current balances? b) Julia buys a book for \(\$15\). What is her new balance? c) Tom deposits \(\$10\). What is his new balance? d) After these changes, who has the greater balance, and what is the difference between the balances?

Find each requested value. a) Evaluate \((+7)-(-15)\). b) What number must be subtracted from \(-3.8\) to get \(-10\)? c) Subtract \(+1 \frac{3}{4}\) from \(-2 \frac{1}{2}\).

Explore how changing the order affects subtraction. Let \(x=-6.2\) and \(y=-3.4\). a) Find \(x-y\). b) Find \(y-x\). c) Compare your answers from a) and b). What do you notice? d) A student claims, “When you subtract one negative number from another negative number, the result is always positive.” Give a counterexample to show that the claim is false.

Evaluate each expression. Write each result as a simplified fraction or mixed number. 1) \(\left(-\frac{7}{8}\right)-\left(+\frac{1}{8}\right)-(-3)\) 2) \(\left(+\frac{2}{5}\right)-\left(-\frac{1}{2}\right)-\left(+\frac{9}{10}\right)\) 3) \(\left(-2 \frac{1}{4}\right)-\left(-\frac{3}{4}\right)-(+1)\)

Let \(a\) and \(b\) be positive rational numbers. a) What numbers must be included with the set of positive rational numbers so that \(a-b\) is always in the resulting set, no matter which positive rational numbers are chosen? b) State the condition on \(a\) and \(b\) that makes \(a-b\): 1. equal to \(0\). 2. negative. 3. positive. c) Give one subtraction example with fractions or decimals for each case in part b).

The expressions are \(A=a-b+c\) and \(B=a-(b+c)\). Evaluate and compare them for each assignment. a) \(a=10\), \(b=-4\), \(c=2\) b) \(a=-3.5\), \(b=1.5\), \(c=-5\)

Evaluate \(T=-x-(y-z)\) for each set of rational numbers. a) \(x=\frac{2}{3}\), \(y=-\frac{1}{6}\), \(z=\frac{1}{2}\) b) \(x=-2\frac{1}{4}\), \(y=1\frac{3}{8}\), \(z=-0.5\)

A weather station recorded these temperatures at midnight during one week: Monday: \(-4\,^\circ\text{C}\), Tuesday: \(-1\,^\circ\text{C}\), Wednesday: \(3\,^\circ\text{C}\), Thursday: \(5\,^\circ\text{C}\), Friday: \(-2\,^\circ\text{C}\), Saturday: \(-6\,^\circ\text{C}\), Sunday: \(0\,^\circ\text{C}\). a) Find the highest and lowest temperatures. b) Find the difference between the warmest and coldest temperatures. c) List the days from coldest to warmest.

Let \(a\) and \(b\) be rational numbers. a) Under what condition is \(a - b > a\)? Explain why and give a numerical example. b) Under what condition does \(a - b = a + b\)? Justify your answer algebraically.

An elevator labels street level as \(0\), levels above it with positive integers, and parking levels below it with negative integers. A passenger enters on Level \(5\) and makes these trips in order: 1. Down \(7\) levels 2. Up \(3\) levels 3. Down \(4\) levels 4. Up \(2\) levels a) On which level does the passenger end? b) What is the lowest level reached during the entire trip? c) How many levels apart are the highest and lowest points of the trip?

Analyze each statement about rational-number operations. a) Can \(x-y\) be greater than \(x\)? Explain or give an example. b) If \(a\ne0\) and \(b\ne0\), can \(a\div b\) equal \(0\)? Explain. c) Is the opposite \(-x\) always less than \(x\)? Consider positive values, negative values, and zero.

Use the number line. a) Write the value of each point \(A\), \(B\), and \(C\) as both a decimal and a fraction in simplest form or a mixed number. b) Find the distance from \(A\) to \(B\) and from \(B\) to \(C\).

Operations with decimals can be shown clearly on a number line. Each arrow shows the direction and size of a jump. Find the numbers represented by the red question marks on the number lines.

These number lines show subtracting positive numbers as jumps to the left. Find the value represented by the question mark in each diagram. a) Where does the jump end? b) Where did the jump begin? c) What is the missing middle value?

The plot shows the temperature at a mountain weather station from \(6{:}00\) a.m. to \(4{:}00\) p.m. on a winter day. Three students wrote different expressions to determine the temperature at \(4{:}00\) p.m. Lucas: \(-3-1+2+3+4+2+0-1-3-4-2\) Clara: \(-3+(2+3+4+2)-(1+1+3+4+2)\) Jonas: \(-3-1+11-10\) a) Evaluate each expression. What temperature does each expression give for \(4{:}00\) p.m.? b) Explain how each student grouped or represented the hourly temperature changes shown in the plot.

The plot shows the air temperature measured at the same time on seven consecutive days at a mountain weather station. Two meteorologists summarize the week: Mr. Frost writes \(T_1=-4+2+3-1-3+2+3\). Ms. Degree writes \(T_2=(2+3+2+3)-(1+3)\). Evaluate both expressions and explain what information each result gives about the weather station.

A hiking group is planning a route through the mountains. The bar graph shows the elevation of each stop in feet. a) Read the elevations of the six stops from the graph. b) Find the elevation change between each pair of consecutive stops. Use a negative sign for a descent. c) Find the sum of all the elevation changes from part b). d) A hiker says, “The finish is only \(300\,\text{ft}\) above the start, so we climb only \(300\,\text{ft}\) in all.” Explain why this statement is incorrect.

Find each missing starting number for the subtraction shown on the number line.

Complete each chain of jumps on the number line. Find every value marked with a question mark.

Use the number line to find the missing starting value.

Use the number line to find the missing intermediate and ending values.

The number-line diagram shows a temperature in the morning, its change by noon, and its change from noon to evening. What is the evening temperature?

The number line shows a temperature change from the previous evening to the morning low. Find the previous evening's temperature.

Let \(A=\frac{5}{8}\) and \(B=\frac{6}{8}\). 1. Find the fraction \(M\) exactly halfway between \(A\) and \(B\). 2. Starting with \(M\), repeatedly find the midpoint between \(A\) and the most recently found midpoint. Explain why this produces infinitely many different fractions between \(\frac{5}{8}\) and \(\frac{6}{8}\).

A student writes \(6 \frac{1}{4}-2 \frac{2}{3}=(6-2)+\left(\frac{2}{3}-\frac{1}{4}\right)=4+\left(\frac{8}{12}-\frac{3}{12}\right)=4 \frac{5}{12}\). Explain the error, then calculate the correct result.

Evaluate each expression. Pay attention to signs and simplify each result. a) \(2 \frac{1}{3}\cdot(4.5-6)\div\frac{7}{4}\) b) \(\frac{5}{12}\div\frac{25}{18}-0.4\cdot1 \frac{1}{2}\)

Divide the six number cards into two groups of three so that the sums of the groups are equal. \(1.2,\ \frac{3}{10},\ -1.5,\ 0.8,\ -0.2,\ -\frac{3}{5}\)

In \(m - s = d\), the minuend \(m\) is increased by \(10\). Determine how the subtrahend \(s\) must change so that the new difference has each property. a) The difference stays the same. b) The difference is \(15\) greater than before. c) The difference is \(5\) less than before.

Decide whether each claim is true or false. Explain. a) If you add twice the opposite of a number to the original number, the result is the opposite of the original number. b) Two different integers can have the same opposite.

Write one subtraction expression with a value of \(-35\) for each condition. a) Both numbers are positive integers. b) The starting number is a negative integer, and the number being subtracted is a positive integer. c) Both numbers are negative integers.

Simon claims, “When I subtract one integer from another integer, the result is always less than the first integer.” Is Simon correct? Use at least two integer examples to justify your answer.

Consider the expressions: (I) \((-48)-(-122)\) (II) \((-210)-(+90)\) a) Rewrite each subtraction as addition and evaluate. b) What number must be subtracted from each answer in part a) to get \(10\)? c) For each answer in part a), what number must be subtracted to get its opposite?

Consider the expressions: (I) \((-15)-(+85)\) (II) \((+40)-(+110)\) a) Rewrite each subtraction as addition and evaluate. b) What number must be subtracted from the value of (I) to get the value of (II)? c) What number must be subtracted from the value of (II) to get the opposite of the value of (I)?

Evaluate each expression one set of parentheses at a time. a) \(145-(34+156-20)-(55+45)\) b) \(-(-760+140)-[(24-110)+480]\)

Consider the expression \(2500 - [750 - (150 - 50)]\). First evaluate the expression. Then determine how its value changes if each of the four numbers is increased by \(10\). Explain why.

Write an expression with grouping symbols for the description, and then evaluate it. From the opposite of the sum of \(-315\) and \(105\), subtract the difference of \(45\) and \(-82\).