Word Problems (Algebra)
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enCreating Expressions, Equations, and Inequalities
https://edboost.org/node/585
<span>Creating Expressions, Equations, and Inequalities</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>One of the most critical skills in algebra is learning to write an equation. Being able to translate a problem -- whether a word problem or a real life problem -- into an equation opens up an entire realm of math.</p><p>So, how do you translate words into an equation?</p><p> </p><p>First, some basics: </p><ul><li><strong>Expressions, equations, and inequalities contain numbers.</strong> If you are given a relevant number, figure out how it fits into the equation.</li><li><strong>Numbers are often unknown.</strong> These unknown numbers should be represented by variables. Variables can be any letter. We often use x and n, but any letters will do. Generally, stay away from variables that look like numbers (e.g., l and g). If you are using several variables, it's good to have letters that are distinct (c and r can get confusing). If you are doing a word problem, it sometimes helps to use letters that represent items (e.g., if the farmer has more wheat then corn, then $w>c$ makes sense).<ul><li>Several words signal that you need to insert a variable:</li></ul></li></ul><table style="margin-left:auto;margin-right:auto;" border="1" cellpadding="2" cellspacing="2" bgcolor="66999FF"><caption><strong>How to know when to insert a variable?</strong></caption><tbody><tr><td style="text-align:center;"><strong>Key Words</strong></td><td style="text-align:center;"><strong>What number...</strong></td><td style="text-align:center;"><strong>A number...</strong></td><td style="text-align:center;"><strong>What quantity</strong></td><td style="text-align:center;"><strong>What amount</strong></td></tr><tr><td style="text-align:center;"><strong>Examples:</strong></td><td style="text-align:center;"><p>What number is equal to 3x and 4x-1?</p><p><br>$n=3x$<br>$n=4x-1$</p></td><td style="text-align:center;"><p>A number is twice as big as 3...</p><p><br>$x=2\times3$</p></td><td style="text-align:center;"><p>What quantity of apples is greater than 3 but less than 7?</p><p><br>$3>x>7$</p></td><td style="text-align:center;"><p>What amount of glue is need to fill 3 4-quart containers?</p><p><br>$g=3\times4$</p></td></tr></tbody></table><p> </p><div> </div><div>And, what makes an equation/inequality solvable? (Note: expressions are not solvable. They just show how variables and numbers are related. Equations and inequalities are solvable.)</div><ul><li>An <strong>equation</strong> contains an equal sign $=$</li><li>An <strong>inequality</strong> is like an equation, but it contains a greater than $>$, less than $<$, greater or equal to $\geq$, or less than or equal to $\leq$ sign. </li><li>An <strong>expression</strong> is set of numbers, variables, and operations but does not have an equal or inequality sign. Expressions can be simplified, but they cannot be solved.</li></ul><p> </p><table border="1" cellpadding="2" cellspacing="2" align="center" bgcolor="6699FF"><caption><strong>How do you know which equal/unequal symbol to use?</strong></caption><tbody><tr><td style="text-align:center;"><strong>Symbol</strong></td><td style="text-align:center;"><strong>Equal $=$</strong></td><td style="text-align:center;"><strong>Greater than $>$</strong></td><td style="text-align:center;"><strong>Less than $<$</strong></td><td style="text-align:center;"><strong>Greater than or equal to $\geq$</strong></td><td style="text-align:center;"><strong>Less than or equal to $\leq$</strong></td></tr><tr><td style="text-align:center;"><strong>Key Words</strong></td><td style="text-align:center;"><p style="text-align:center;">equals<br>is equal to<br>is the same as<br>is congruent to<br>is<br>the answer is</p></td><td style="text-align:center;">is greater than<br>is more than<br>is larger than<br>>exceeds</td><td style="text-align:center;">is less than<br>is lower than<br>is smaller than</td><td style="text-align:center;">is greater than or equal to<br>is at least<br>is no less than</td><td style="text-align:center;"><p style="text-align:center;">is less than or equal to<br>is at most<br>is no more than </p></td></tr><tr><td style="text-align:center;"><strong>Examples</strong></td><td style="text-align:center;"><p>A number is the same as 6:</p><p> $n\mathbf{ = } 6$</p></td><td style="text-align:center;"><p>x is more than y:</p><p>$x\mathbf{>}3+y$</p></td><td style="text-align:center;"><p>g is less then 10:</p><p>$g\mathbf{<}10$</p></td><td style="text-align:center;"><p>x is no less than 64:</p><p>$x\mathbf{\geq}64$</p></td><td><p style="text-align:center;">He earns no more than \$20/hour</p><p style="text-align:center;">$m\mathbf{\leq}20$</p></td></tr></tbody></table><p> </p><p>And finally, what really defines an expression/equation/inequality are the operations. Algebra operations are the same operations you're used to using in math every day: addition $+$, subtraction $-$, multiplication $\times$, and division $\div$.</p><p> </p><table border="1" cellpadding="2" cellspacing="2" align="center" bgcolor="6699FF"><caption><strong>What operations do you use?</strong></caption><tbody><tr><td style="text-align:center;"><strong>Symbol</strong></td><td style="text-align:center;"><strong>Addition $+$</strong></td><td style="text-align:center;"><strong>Subtraction $-$</strong></td><td style="text-align:center;"><strong>Multiplication $\times$</strong></td><td style="text-align:center;"><strong>Division $\div$</strong></td></tr><tr><td style="text-align:center;"><strong>Key Words</strong></td><td style="text-align:center;"><p style="text-align:center;">add<br>sum<br>plus<br>in addition to<br>all together<br>in total<br>and<br>combined<br>more<br>increased by</p></td><td style="text-align:center;">subtract<br>minus<br>less<br>take away<br>difference<br>change<br>left over<br>decreased by<br>reduced by</td><td style="text-align:center;"><p>multiplied by<br>twice<br>times<br>product<br>by<br>of </p><p> </p></td><td style="text-align:center;">divided by<br>quotient<br>per<br>each</td></tr><tr><td style="text-align:center;"><strong>Examples</strong></td><td style="text-align:center;"><p>A number plus 6 is 8</p><p> $n+6=8$</p><p> </p></td><td style="text-align:center;"><p>The difference between 5 and x is 3</p><p>$5-x=3$</p><p>7 is 12 less than x</p><p>$x-12=7$</p></td><td style="text-align:center;"><p>The product of 5 and x is 20</p><p>$5\times x=20$</p><p> </p></td><td style="text-align:center;"><p>A number divided into 6 equal groups is 2</p><p>$x\div6=2$</p></td></tr><tr><td style="text-align:center;"><strong>Special notes:</strong></td><td style="text-align:center;">You can add numbers in any order (because of the communicative property of addition).</td><td style="text-align:center;">Order matters! It's often switched. So, x less 7 is $x-7$ but 7 less than x is also $x-7$. Think about it: You have \$10. Your friend has three dollars less than you, would you do $3-10$ or $10-3$? Always think about order when you write a subtraction problem. </td><td style="text-align:center;"><p>You can multiple numbers in any order (because of the communicative property of multiplication).</p><p>"Of" is often used when multiplying by decimals or fractions: one half of 20is ten or 45% of 100 is 45. In both of these cases "of" means multiply.</p></td><td style="text-align:center;">When you write a division problem either with a division sign ($19\div4$) you enter numbers into a calculator in that order. When you write a division problem as a fraction ($\dfrac{19}{3}$), you enter those numbers into a calculator from top to bottom ($19\div3$). But, if you divide on scratch paper, using the "house," you write numbers in the opposite order: $3\vert\overline{19}$.</td></tr></tbody></table><p> </p><p>Once you have your numbers, variables, operators, and equal or inequality signs, you have everything you need to put together an expression/equation/inequality (or several!).</p><p>Remember, you can piece together an equation (or set of equations) even if you're not entirely sure what you're doing. One of the reasons we write math equations is to make problems clearer. Writing math in words can be clumsy and imprecise. So, let's use key words above to piece together the following equations:</p><p style="padding-left:30px;"><em>Example:</em> <strong>Write an equation that shows the relationship: 7 less than x is equal to the product of x and 2.</strong></p><p style="padding-left:30px;">Take it step by step: </p><p style="padding-left:30px;">$$\eqalign{\text{7 less than x }\rightarrow x-7&\text{}&&\mathbf{ Less\; than}\text{ are subtraction words (remember, it's one of the subtraction} \\&\text{}&&\text{words where you have to think about order).}\\\text{Next step: is equal to }\rightarrow x-7&=&&\mathbf{Is \;equal\; to}\text{ means equal sign.}\\\text{Final step: product of x and 2 }\rightarrow x-7&=x\times2&&\mathbf{Product}\text{ is a multiplication word and order does not matter}\\&\text{}&&\text{ in multiplication.}\\x-7&=2x&&\text{Usually in math we show multiplication just by putting numbers}\\&\text{}&&\text{and/or variables attached to each other.}}$$</p><p style="padding-left:60px;"> </p><p><span style="text-align:center;">The one other complicating factor when writing equations is order of operations. If you need to review order of operations see the lesson </span><a href="4447"><span style="text-align:center;">Order of Operations</span></a><span style="text-align:center;">. Equations follow standard math order of operations. The only way to change that is to use parentheses or brackets. Pay attention when a word problem seems to be grouping numbers or variables. </span></p><p><br> </p><p style="padding-left:30px;"><em>Example:</em> <strong>Fifteen divided by 4 is greater than twice the sum of 13 and 4.</strong></p><p style="padding-left:30px;">Take it step by step: </p><p style="padding-left:30px;">$$\eqalign{\text{15 divided by 4 }\rightarrow 15\div 4&\text{}&&\text{}\mathbf{ Divided\; by}\text{ are division words}\\\text{Next step: is greater than }\rightarrow 15\div 4&>&&\text{}\mathbf{ Is\; greater\; than}\text{ means an inequality with the "mouth"} \\&\text{}&&\text{of the inequality sign facing the greater side.}\\\text{Final step: twice the sum of 13 and 5 }\rightarrow 15\div 4& > 2\times &&\text{}\mathbf{ Twice}\text{ is a multiplication word that means multiply by 2.}\\15\div 4&> 2\times (13+4)\text{}&&\mathbf{ Sum}\text{ is an addition word. The way that this is written suggests that }\\\text{}&\text{}&&\text{ it's the sum that isbeing multiplied, not just 13 or 4,}\\&\text{}&&\text{ so group with parentheses.}}$$</p><p style="padding-left:30px;"> </p><p style="text-align:left;">Overall, writing equations takes practice (see the lesson on <a href="http://www.boostbase.org/math/creating-equations-word-problems">Creating Equations from Word Problems</a> to take this to the next level). But once you memorize the key words and get familiar with the little tricks (like "less than" usually requiring you to switch the order of the numbers), you can write almost any kind of equation, including equations that are embedded in word problems. </p></div></div>
<span><span>edboost</span></span>
<span><time datetime="2024-06-19T12:51:40-07:00" title="Wednesday, June 19, 2024 - 12:51">Wed, 06/19/2024 - 12:51</time>
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<h4><i class="icon-bookmark"></i> Practice Problems:</h4>
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<li><article data-history-node-id="403" class="node node-type-math-practice-problems node-view-mode-default">
<h2 class="node-title">
<a href="https://edboost.org/index.php/node/403" rel="bookmark"><span>Reading and Writing Expressions</span>
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<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>Translate the following into numerical expressions:</p><ol><li>The sum of a number, eight, and seven.</li><li>Three times the sum of a number and nine.</li><li>Four less than four times the difference of ten and a number.</li><li>The mean of a number, five, seven, and twelve.</li><li>Six times the quotient of five and a number is greater than two.</li><li>The difference of eight and a number, divided by four equals the sum of the number and two.</li><li>The difference of two sums: eight and a number and six and another number is at least the sum of the second number and one.</li><li>The sum of eight times a number and three times the same number is less than the difference of the number and ten.</li><li>The difference of eleven times a number and two times another number equals twenty less than the first number.</li><li>The total of six times a number and the same number divided by three is no more than four times the number.</li></ol><p>Translate the following expressions, equations, and inequalities into words:</p><ol start="11"><li>$3x+9$</li><li>$3(2x+1)$</li><li>$\dfrac{4}{x}+9x$</li><li>$\dfrac{5+x}{2}\leq 42$</li><li>$(3x+7)-(2x+2)>6x$</li><li>$x(5+x)-3<60$</li><li>$x-9\geq 6x$</li><li>$\dfrac{x}{6+x}=6x-y$</li><li>$5x+6y>\dfrac{x}{5y}$</li><li>$5x(\dfrac{x}{6})=7+x$</li></ol></div></div>
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<h4><i class="icon-bookmark"></i> Skill:</h4>
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<li><a href="https://edboost.org/taxonomy/term/57" hreflang="en">Word Problems (Algebra)</a></li>
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<h4><i class="icon-bookmark"></i> Common Core Grade Level/Subject</h4>
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<li><a href="https://edboost.org/taxonomy/term/16" hreflang="en">Algebra I</a></li>
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<h4><i class="icon-bookmark"></i> EdBoost Test:</h4>
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<li><a href="https://edboost.org/taxonomy/term/4" hreflang="en">Algebra I</a></li>
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Wed, 19 Jun 2024 19:51:40 +0000edboost585 at https://edboost.orgCreating Equations (Unknowns in terms of same variable)
https://edboost.org/node/584
<span>Creating Equations (Unknowns in terms of same variable)</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>In algebra we use variables to represent unknowns. And, by using the tips and skills in the Creating Equations lesson, we can put together an equation from most word problems. However, it's difficult (and sometimes impossible) to solve a single equation with two variables. So, how can we take an equation with two unknowns and write an equation with only one variable?</p><p>Often word problems tell you how the two unknowns are related to each other, giving you a way to write one uknown "in terms of" another unknown.</p><p>When we say "in terms of" it just means that we are using one variable to define another. Let's try a problem:</p><p><em>Example</em>: Avery is 4 years older than Kielan. If you add their ages together, they are 16 years old. How old is Avery?</p><p>Let's define our variables:</p><p>Avery's age = $a$</p><p>Kielan's age = $k$</p><p>$a + k = 16$</p><p>But, with two variables, we will never know exactly what age either Avery and Kielan are.</p><p>BUT! The problem gives us more information: "Avery is 4 years older than Kielan." That means that we can give Avery's age in terms of Kielan's age (remember, that just means that we will use Kielan's variable (k) in Avery's variable:</p><p>Avery's age $= a = k+4$</p><p>Kielan's age $= k$</p><p>Now, we can take our original equation and substitute the term $(k-4)$ for the variable $a$ (remember, if things are equal, you can swap them in and out of equations -- they are equal so using a new equal form does not change the equation).</p><p>$$\eqalign{a+k&=16\\(k+4)+k&=16}$$</p><p>From here, we combine like terms and solve the equation:</p><p> </p><p>$$\eqalign{(k+4)+k&=16\\2k+4&=16\\-4\text{}&\text{}-4\\2k&=12\\\div2\text{}&\text{}\div2\\k&=6}$$</p><p>So, Kielan is six years old (and if $a=k+4\text{, then } a=10$) and Avery is 10 years old.</p><p>All we did there was write our equation, write one variable in terms of the other, and then substitute our new variable (with the k in it) for the old variable (a). Then we solve just like we would any algebra equation.</p><p> </p><p>This strategy works for all kinds of problems with two, related unknowns. Let's try another.</p><p><em>Example</em>: You and a friend rent a car together. However, because you are the one who wanted to get a fancy car, you pay three times as much as your friend. If the rental car cost \$78, how much did you pay toward the rental?</p><p>Let's define our variables:</p><p>Your amount = $y$</p><p>Your friend's amount = $f$</p><p>$y + f = 78$</p><p>Again, with two variables, its hard to know who paid how much.</p><p>But, the problem gives more information: "..you pay three times as much as your friend." That means that we can give the amount you paid (y) in terms of the amount your friend paid (f).</p><p>Your amount $= y = 3\times f = 3f$</p><p>Your friend's amount $= f$</p><p>Now, take the original equation and substitute the term $(3f)$ for the variable $y$.</p><p>$$\eqalign{y+f&=78\\(3f)+f&=78}$$</p><p>From here, combine like terms and solve the equation:</p><p> </p><p>$$\eqalign{(3f)+f&=78\\4f&=78\\\div4\text{}&\text{}\div4\\f&=19.5}$$</p><p>$f=19.50$, so, your friend paid \$19.50. What you paid is equal to $3f$, so you paid: \$58.50.</p><p> </p><p>Overall, when faced with a word problem with two unknowns, see if the unknowns are related. If they are, you can write one unknown in terms of the other, and then just solve for one variable. Be careful when writing your final answer: the problem could ask for either the variable or the other term. So, like this last example, you might have to do some math, using the variable you know, to solve for the other variable. </p></div></div>
<span><span>edboost</span></span>
<span><time datetime="2024-06-19T12:50:05-07:00" title="Wednesday, June 19, 2024 - 12:50">Wed, 06/19/2024 - 12:50</time>
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<div class="node-taxonomy-container field--name-field-practice-problems field--type-entity-reference field--label-above">
<h4><i class="icon-bookmark"></i> Practice Problems:</h4>
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<li><article data-history-node-id="298" class="node node-type-math-practice-problems node-view-mode-default">
<h2 class="node-title">
<a href="https://edboost.org/index.php/node/298" rel="bookmark"><span>Create Equations (Unknowns in terms of the same variable)</span>
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<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>Write and solve equations for the word problems:</p><ol><li>You go to a burger joint. A burger costs x. Fries cost .45 less than burgers. If you order 2 burgers and 3 orders of fries, and spend \$17.90, how much does a burger cost? How much do fries cost?</li><li>At a competing burger joint, drinks cost 50 cents less than fries. Burgers cost twice as much as fries. If you order 1 burger, one order of fries, and one drink, it costs \$7.70. How much does a burger cost? How much do fries cost? How much do drinks cost?</li><li>James and Richie took a math test. Richie scored twice as many points as James did. Together, they got 117 points. How many points did Richie score? How many points did James score?</li><li>James, Richie, and Naseeha all took the same science test. James score 3 times as many points as Richie did. Nasheeha scored 4 more points than James did. All together, they scored 235 points. How many points did James get? How many points did Richie get? How many points did Naseeha get?</li><li>Bella and Mimi collect trading cards. Mimi has 5 fewer than 3 times as many cards as Bella does. If Mimi has 47 cards, how many cards does Bella have? </li><li>Khizar and Umar are playing basketball. Umar scores 6 more than 4 times as many points as Khizar does. Together they scored 76 points. How many points does Khizar score? How many points does Umar score?</li><li>Anthony, Dylan, and Kevin are working on a group project together. Anthony answers half as many questions as Kevin does. Kevin answers 8 fewer questions than Dylan does. If there are 38 questions on the project. How many did Anthony, Kevin, and Dylan each answer?</li><li>After they finished their project, Anthony, Dylan, and Kevin went out to get lunch. They each bought a sandwich. Dylan's sandwich cost \$1 more than Kevin's sandwich. Anthony's sandwich cost 85 cents less than Kevin's sandwich. They each got a drink for \$2 each. If they split the cost evenly, and each paid \$7.05, how much did each boy's sandwich cost?</li></ol></div></div>
<div class="node-taxonomy-container field--name-field-answer-key field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Answer Key:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/index.php/node/85" hreflang="en">Create Equations (Unknowns in terms of the same variable)</a></li>
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<h4><i class="icon-bookmark"></i> Skill:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/index.php/taxonomy/term/57" hreflang="en">Word Problems (Algebra)</a></li>
</ul>
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<div class="node-taxonomy-container field--name-field-common-core-grade-level-su field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Common Core Grade Level/Subject</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/index.php/taxonomy/term/16" hreflang="en">Algebra I</a></li>
</ul>
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Wed, 19 Jun 2024 19:50:05 +0000edboost584 at https://edboost.orgCreate Equations from Word Problems
https://edboost.org/node/583
<span>Create Equations from Word Problems</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>One of the algebra skills that students struggle with the most is writing equations from word problems. Ironically, translating real life problems into math is one of the key ways that algebra can become useful in real life (All those times you ask yourself, why do I need to learn this? This is why you need to learn algebra!).</p><p>If you need to practice putting equations togther, visit the <a href="4314" target="_blank">Creating Equations Lesson</a> for all of the details. But, if you can create equations when you are giving the numbers, you're ready to try it when you're given words.</p><p>Below is a quick review of the key words that signal what types of math operations that a word problem is asking for. We've included some word problem examples as well.</p><table cellspacing="2" cellpadding="2" border="1" bgcolor="6699FF" align="center"><caption><strong>What operations do you use?</strong></caption><tbody><tr><td style="text-align:center;"><strong>Symbol</strong></td><td style="text-align:center;"><strong>Addition $+$</strong></td><td style="text-align:center;"><strong>Subtraction $-$</strong></td><td style="text-align:center;"><strong>Multiplication $\times$</strong></td><td style="text-align:center;"><strong>Division $\div$</strong></td></tr><tr><td style="text-align:center;"><strong>Key Words</strong></td><td style="text-align:center;"><p style="text-align:center;">add<br>sum<br>plus<br>in addition to<br>all together<br>in total<br>and<br>combined<br>more<br>increased by</p></td><td style="text-align:center;">subtract<br>minus<br>less<br>take away<br>difference<br>change<br>left over<br>decreased by<br>reduced by</td><td style="text-align:center;"><p>multiplied by<br>twice<br>times<br>product<br>by<br>of </p><p> </p></td><td style="text-align:center;">divided by<br>quotient<br>per<br>each</td></tr><tr><td style="text-align:center;"><strong>Examples</strong></td><td style="text-align:center;"><p>A number plus 6 is 8</p><p> $n+6=8$</p><p>If a ball costs \$6 and a bat costs \$12, what do they call all together?</p><p>$6+12=$</p></td><td style="text-align:center;"><p>The difference between 5 and x is 3</p><p>$5-x=3$</p><p>7 is 12 less than x</p><p>$x-12=7$</p></td><td style="text-align:center;"><p>The product of 5 and x is 20</p><p>$5\times x=20$</p><p>He scored 3 times as many as Abe, who scored 14.</p><p>$3\times14=\text{?}$</p></td><td style="text-align:center;"><p>If you divide 6 apples among 2 kids, who many apples does each kid get?</p><p>$6\div2=\text{?}$</p></td></tr><tr><td style="text-align:center;"><strong>Special notes:</strong></td><td style="text-align:center;">You can add numbers in any order (because of the communicative property of addition).</td><td style="text-align:center;">Order matters! It's often switched. So, x less 7 is $x-7$ but 7 less than x is also $x-7$. Think about it: You have \$10. Your friend has three dollars less than you, would you do $3-10$ or $10-3$? Always think about order when you write a subtraction problem. </td><td style="text-align:center;"><p>You can multiply numbers in any order (because of the communicative property of multiplication).</p><p>"Of" is often used when multiplying by decimals or fractions: one half of 20is ten or 45% of 100 is 45. In both of these cases "of" means multiply.</p></td><td style="text-align:center;">When you write a division problem either with a division sign ($19\div4$) you enter numbers into a calculator in that order. When you write a division problem as a fraction ($\dfrac{19}{3}$), you enter those numbers into a calculator from top to bottom ($19\div3$). But, if you divide on scratch paper, using the "house," you write numbers in the opposite order: $3\vert\overline{19}$.</td></tr></tbody></table><p> </p><p>Once you have these key words in your head, you're ready to read the questions and start to break it down into an equation.</p><p style="padding-left:30px;"><em>Example:</em> <strong>The five members of a basketball team are getting new practice equipment. Shoes cost \$129 a pair. Balls cost \$15 each. Uniforms are \$80 a piece. What is the total cost of all of the new equipment for the team?</strong></p><p style="padding-left:30px;">Sometimes the easiest thing to do is write out words and operations (don't worry about numbers and variables yet).</p><p style="padding-left:30px;">You are finding a <strong>total</strong> so you are going to want to add.</p><p style="padding-left:60px;">$\text{shoes} + \text{balls} + \text{uniforms} = \text{total}$</p><p style="padding-left:30px;">Great. But, remember that there are five players on the team.</p><p style="padding-left:60px;">$5\times\text{shoes} + 5\times\text{balls} + 5\times \text{uniforms} = \text{total}$</p><p style="padding-left:30px;">That makes sense, right? Now, just replace your words with the numbers provided in the word problem:</p><p style="padding-left:60px;">$5\times 129 + 5\times 15 + 5\times 80= \text{total}$</p><p style="padding-left:30px;">Now, you can solve:</p><p style="padding-left:60px;">$$\eqalign{5\times 129 + 5\times 15 + 5\times 80 &= \text{total}\\645 + 75 + 400 &=\text{total}\\1120 &=\text{total}}$$</p><p style="padding-left:60px;"> </p><p>Sometimes, a word problem will give you an answer and you have to write an equation in order to work backwards and solve for some other unknown number.</p><p style="padding-left:30px;"><em>Example:</em> <strong>Abraham purchased a gaming system for \$299 and several games at \$39.99 each. If he paid \$418.97, how many games did he purchase?</strong></p><p style="padding-left:30px;">Again, the easiest thing to do is write out words and operations (don't worry about numbers and variables yet).</p><p style="padding-left:30px;">You are finding a <strong>total</strong> so you are going to want to add. </p><p style="padding-left:60px;">$\text{game system} + \text{games} = \text{total}$</p><p style="padding-left:30px;">But, remember that there are several games -- but we don't know how many, let's use a variable.</p><p style="padding-left:60px;">$\text{game system} + g\times\text{games} = \text{total}$</p><p style="padding-left:30px;">Now, replace your words with the numbers provided in the word problem:</p><p style="padding-left:60px;">$299 + g\times39.99 = 418.97$</p><p style="padding-left:30px;">Now, you can solve:</p><p style="padding-left:60px;">$$\eqalign{$299 + g\times39.99 &= &&418.97\\299 + 39.99g &= &&418.97&&&\text{Rewrite in algebra format}\\-299& &&-299 &&&\text{Subtract 299 from each side}\\39.99g& = &&119.97\\\div 39.99& && \div 39.99&&&\text{divide each side by 39.99 to get } g \text{ alone}\\g&=&&3777&&&\text{He bought 3 games}}$$</p><p> </p><p>The most complicated word problems have a lot of pieces, sometimes multiple variables, sometimes just a lot of terms. Move them them slowly and logically and you'll put the math problem together. </p><p style="padding-left:30px;text-align:left;"><em>Example</em>: <strong>A teacher has some number of cookies. She is going to order more cookies so that she doubles her stock, plus an additional 30. There are 24 kids in her class when she distributes all of the cookies, we know each student gets 3. Write an equation to show how many cookies the teacher started with.</strong></p><p style="padding-left:60px;text-align:left;">There's a lot going on here. First we need to figure out which quantity is unknown. Then we have to add in the new order. Then we have to divide the cookies by students. In the end, we set it equal to 3 (because each student gets three cookies). Let's set this one up, step by step, and then solve it to see how we can actually figure out the answer to this problem.</p><p style="padding-left:60px;text-align:left;">$$\eqalign{&x \text{ cookies}&&\text{ Teacher starts with some number of cookies. We use a variable. }\\&x\times 2&&\text{ Double the number of cookies}\\&2x + 30&&\text{ Simplify to 2x then add another 30 cookies}\\&\dfrac{2x+30}{24}=&&\text{ Divide total cookies by each student (24)}\\&\dfrac{2x+30}{24}=3&&\text{ Set expression equal to 3. All cookies, divided by 24 students = 3 cookies each}}$$</p><p style="padding-left:60px;text-align:left;">Now that we have our equation, we can use algebra to solve for x and find out how many cookies the teacher started with. </p><p style="padding-left:60px;text-align:left;">$$\eqalign{&\dfrac{2x+30}{24}&&=&&&3&&&&\text{Our equation. Variable x represents number of cookies teacher started with}\\&\small{\times 24}&&\text{}&&&\small{\times 24}&&&&\text{ Multiply each side of the equation by 24 to clear the fraction}\\&2x+30&&=&&&72\\&\small{-30}&&\text{}&&&\small{-30}&&&&\text{ Add 30 to each side to clear the addition term}\\&2x&&=&&&102\\&\small{\div 2}&&\text{}&&&\small{\div 2}&&&&\text{ Divide each side by 2 to clear the 2 and isolate x}\\&x&&=&&&51\text{ cookies}}$$</p><p style="text-align:left;"> </p><p style="text-align:left;">Note that sometimes, a word problem does not give you an answer. You may not be able to solve this problem: you just need to write the expression that represents the word problem.</p><p style="padding-left:30px;text-align:left;"><em>Example</em>: <strong>Games4U has 35 video games in stock. They then receive a shipment of 6y more games. But, they sell 3 games. What is the total number of games that they have in stock now?</strong></p><p style="padding-left:60px;text-align:left;">The key word "total" signals that you're going to want to add. But they also sell some games (so, will that be an addition or a reduction?). Let's start from the beginning.</p><p style="padding-left:60px;text-align:left;">$$\eqalign{&35 \text{ games}&&\text{ They start with 35 games in stock}\\+&6y&&\text{ Add the new shipment}\\-&3&&\text{ Subtract the games that they sold}\\&35+6y-3=\text{total}&&\text{ Final expression}}$$</p><p style="padding-left:60px;text-align:left;"> </p><p style="text-align:left;">The keys to writing equations are:</p><ul><li><strong>Pay attention</strong> to all of the steps, </li><li>Look for <strong>key words</strong>, and </li><li>Use your <strong>logic</strong>. </li><li>Equations are just math problems in which you do not know all of the numbers. Sometimes it helps to <strong>put real numbers in place of variables just to think about how you would solve the problem if you knew everything</strong> (so, for instance, if you knew that a teacher had 20 cookies, doubled that number, then added 30, what would you do? You would do $20\times 2 + 30$. To convert that same idea to an equation, just replace the $20$ with an $x$ and you're all set). </li><li>The final key to equation success is <strong>practice</strong>. Translating thoughts into math is like learning a new language. The more you practice, the more natural it gets. Did you know that scientists, economists, and other people who work with math a lot often take notes in equation form rather than in word form? They are just so used to thinking in math that they find it easier and simpler than writing things down in words. </li></ul><p style="text-align:left;">So, don't be scared of these kinds of problems. Do the practice, and keep doing more practice whenever you find it, and you'll soon be good at "thinking in math" and creating equations. </p></div></div>
<span><span>edboost</span></span>
<span><time datetime="2024-06-19T12:49:06-07:00" title="Wednesday, June 19, 2024 - 12:49">Wed, 06/19/2024 - 12:49</time>
</span>
<div class="node-taxonomy-container field--name-field-skill field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Skill:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/taxonomy/term/57" hreflang="en">Word Problems (Algebra)</a></li>
</ul>
</div>
<div class="node-taxonomy-container field--name-field-common-core-grade-level-su field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Common Core Grade Level/Subject</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/taxonomy/term/16" hreflang="en">Algebra I</a></li>
</ul>
</div>
<div class="node-taxonomy-container field--name-field-edboost-test field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> EdBoost Test:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/taxonomy/term/4" hreflang="en">Algebra I</a></li>
</ul>
</div>
Wed, 19 Jun 2024 19:49:06 +0000edboost583 at https://edboost.orgMath Logic and Vocabulary
https://edboost.org/node/582
<span>Math Logic and Vocabulary</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>Math is a precise practice and getting math right often relies upon a mutual understanding. When I say "integer," it's important that everyone understand what an integer is (it's any non-decimal number, positive, negative, or 0).</p><p>So, not only is math vocabulary critical to executing math problems, a lot of math tests ask questions that rely explicitly on vocabulary. So, make sure that you know these terms:</p><ul><li><strong>Integers</strong>: Positive or negative <em>whole</em> numbers (including 0), but no decimals or fractions. <em>-96, -21, -2, -1, 0, 1, 2, 56, 121 are all integers.</em></li><li><strong>Consecutive</strong>: In a row. 1, 2, 3 are consecutive numbers. 8, 10, 12 are consecutive even numbers. Consecutive numbers need to occur in a row but can start anywhere on the number line.</li><li><strong>Divisible</strong>: Can be divided by (e.g., 20 is divisible by 1, 2, 4, 5, and 10).</li><li><strong>Remainder</strong>: The number left over after you divide a number. So, if you do 23 divided by 5, the answer is 4 remainder 3 (5 goes in 4 times, with 3 left over). Most calculators will not do remainders (they convert remainders to decimals). The easiest way to do remainder problems is to divide by hand like you did in 5th grade (answers will look like: 4 R 3)</li><li><strong>Even</strong>: Divisible by 2. 2, 4, 6, 8, 10 are even numbers.</li><li><strong>Odd</strong>: Not divisible by 2. 1, 3, 5, 7 are odd numbers.</li><li><strong>Prime</strong>: Divisible only by 1 and itself. 2, 3, 11, 13, 17, 23 are prime numbers. 2 is the only even prime; <strong>1 and 0 are NOT prime numbers </strong>(there is some debate in the math community about this, but according to SAT and ACT, 1 and 0 are not prime, so don’t count them if you are counting prime numbers!).</li><li><strong>Sum</strong>: Answer to an addition problem. The sum of 3 and 4 is 7.</li><li><strong>Difference</strong>: Answer to a subtraction problem. The difference of 10 and 4 is 6.</li><li><strong>Product</strong>: Answer to a multiplication problem. The product of 2 and 4 is 8.</li><li><strong>Quotient</strong>: Answer to a division problem. The quotient of 18 and 9 is 2.</li><li><strong>Factor</strong>: Numbers in a multiplication problem. In the problem 3 x 2 = 6, 3 and 2 are factors. Sometimes SAT uses the word factors to describe all of the numbers that multiply to make another number (or, another way to say it is, all the numbers that a number can be divided by evenly). The factors of 10 are: 1, 2, 5, 10.</li><li><strong>Divisor</strong>: In division, the number divided into the other number (the “divided by” number). If the problem 10 ÷ 2 = 5, 2 is the divisor.</li><li><strong>Dividend</strong>: In division, the number that gets divided. If the problem 10 ÷ 2 = 5, 10 is the dividend.</li><li><strong>Exponent</strong>: The little number next to a base number that means to multiply the number by itself that number of times (e.g., 22 = 2 x 2 = 4; 24 = 2 x 2 x 2 x2 = 16). In 24, 4 is the exponent.</li><li><strong>Base</strong>: The big number next to the exponent in an exponent. In 24, 2 is the base.</li><li><strong>Inclusive</strong>: Including. If a problem says "consider the numbers 6 to 12, inclusive" consider 6, 7, 8, 9, 10, 11, and 12.</li><li><strong>Exclusive</strong>: Excluding. If a problem says "consider the numbers 6 to 12, exclusive" consider only 7, 8, 9, 10, and 11, not the numbers named in the problem.</li><li><p><strong>Coordinate Plane</strong>: The plane, defined by an $x$-axis (horizontal) and $y$-axis (vertical), on which we plot coordinates.</p><img data-entity-uuid="19219a66-3bc5-447e-b4bb-8bff0b9f1650" data-entity-type="file" src="https://edboost.org/sites/default/files/inline-images/coordinateplanequadrants.gif" width="374" height="374" alt="Coordinate Plane with Labeled Quadrants" class="align-center" loading="lazy"><ul><li><strong>Coordinates</strong>: The (x,y) values of a point, that allow you to plot a point on a coordinate plane. These x and y values can be substituted in for x and y in an equation.</li><li><strong>Axis/Axes</strong>: The x and y (or labeled with other letters) lines that define the plane. Sometimes these are in the middle of the plane (when there are negative values on the axes) and sometimes they define the bottom and left sides of the plane (more typical in a graph with only positive values). </li><li><strong>Quadrants</strong>: In a coordinate plane with positive and negative x and y values, there are 4 quadrants, named with roman numerals.</li><li><strong>Intercept</strong>: The point where a line crosses an axis. The y-intercept, where the line crosses the y-axis, is $b$ in the $y=mx+b$ linear equation.</li></ul></li><li><strong>Minimum</strong>: The highest number in a dataset (can also be seen as the highest point on a graph).</li><li><strong>Maximum</strong>: The lowest number in a dataset (can also be seen as the lowest point on a graph).</li></ul><p>Once you know these terms solidly, you'll be able to interpret even tricky math logic and vocabulary questions. </p><p><em>Example</em>: What is the greatest of three consecutive integers whose sum is 24?</p><p>First, define the words in the problem:</p><p><strong>Consecutive</strong> = "in a row"</p><p><strong>Sum</strong> = "answer to an addition problem"</p><p>So, you need three numbers in a row that add up to 24.</p><p>Start with an estimate:</p><p>You need three numbers, in a row that add up to 24. So, let's figure out what number, added up three times, will make a sum of 24: $24 \div 3 = 8$</p><p>So, we know that $8+8+8=24$</p><p>What consecutive numbers add up to 24? </p><p>How about: $7+8+9=$?</p><p>Check your estimate:</p><p>$7+8+9=24$</p><p>Remind yourself what the question wants: "the greatest of three consecutive integers."</p><p>The greatest number of 7, 8, and 9 is <strong>9</strong>.</p><p>Overall, knowing math vocabulary well will help you quickly understand many math problems (and also help you eliminate answers -- if an answer has to be an integer, you can eliminate 2.5 right off the bat!). Knowing math vocabulary, and applying some logic, will also help you work your way through a number of seemingly tricky math problems and puzzles.</p><p>When it doubt, follow the vocab and the rules, make estimates, and guess and check. You may have to guess and check a few times, but do it a few times. Either the answer or a pattern usually becomes clear pretty quickly!</p><p><br> </p></div></div>
<span><span>edboost</span></span>
<span><time datetime="2024-06-19T12:47:27-07:00" title="Wednesday, June 19, 2024 - 12:47">Wed, 06/19/2024 - 12:47</time>
</span>
<div class="node-taxonomy-container field--name-field-practice-problems field--type-entity-reference field--label-above">
<h4><i class="icon-bookmark"></i> Practice Problems:</h4>
<ul class="taxonomy-terms">
<li><article data-history-node-id="397" class="node node-type-math-practice-problems node-view-mode-default">
<h2 class="node-title">
<a href="https://edboost.org/node/397" rel="bookmark"><span>Math Logic and Vocabulary</span>
</a>
</h2>
<div class="node-content">
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>1. What is the greatest integer less than 7?</p><p>2. What is the least integer greater than 6?</p><p>3. What is the greatest integer less than 55?</p><p>4. What is the greatest integer less than 12?</p><p>5. What is the least integer greater than 2?</p><p>6. What is the least integer greater than 0?</p><p>7. If the sum of 3 consecutive integers is 12, what is the greatest of the integers?</p><p>8. If the sum of 5 consecutive integers is 35, what is the least of the integers?</p><p>9. If the sum of 3 consecutive even integers is 30, what is the greatest of the integers?</p><p>10. If the sum of 3 consecutive odd integers is 21, what is the least of the integers?</p><p>11. Which of the following is not a multiple of 6?</p><p>a. 3<br>b. 12<br>c. 24<br>d. 72</p><p>12. Which of the following is a prime factor of 60?</p><p>a. 1<br>b. 5<br>c. 6<br>d. 12</p></div></div>
<div class="node-taxonomy-container field--name-field-answer-key field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Answer Key:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/node/186" hreflang="en">Math Logic and Vocabulary AK</a></li>
</ul>
</div>
</div>
</article>
</li>
</ul>
</div>
<div class="node-taxonomy-container field--name-field-skill field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Skill:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/index.php/taxonomy/term/57" hreflang="en">Word Problems (Algebra)</a></li>
</ul>
</div>
<div class="node-taxonomy-container field--name-field-common-core-grade-level-su field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Common Core Grade Level/Subject</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/index.php/taxonomy/term/16" hreflang="en">Algebra I</a></li>
</ul>
</div>
Wed, 19 Jun 2024 19:47:27 +0000edboost582 at https://edboost.orgInterpret Equations
https://edboost.org/node/581
<span>Interpret Equations</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p><strong>Equations are mathematical sentences.</strong> We write equations to solve for variables that we don't know, but can predict based upon other variables. Some of the most useful -- and difficult -- math problems ask students to write or interpret equations. What does this variable mean? What happens to variable $n$ when variable $x$ goes up or down?</p><p>Equations based on real life (or on word problems that simulate real life) are number sentences that represent real situations. One of the best way to interpret these equations is to break the math sentence apart and turn it back into the real life situation it represents. From there, it's often easy to figure out what variables represent what, and what happens to one variable when another increases or decreases. </p><p>Sometimes the best way to solve this kind of problem is to <strong>think logically about how you would set up an equation</strong> (not using variables, just using the ideas from the question) and then translating that into variables.</p><p><em>Let's try an example:</em></p><p>Kaitlin has a summer job writing reading curriculum based on the new reading standards. The number of standards she still has to write curriculum for can be estimated with the equation $S=107-3x$, where $S$ represents the number of standards she still needs to write curriculum for and $x$ represents the number of days she has worked so far this summer. </p><p><em>What is the meaning of 3 in this equation?</em></p><p><em>What is the meaning of 107 in this equation?</em></p><p><em>What happens to $S$ when $x$ increases?</em></p><p> </p><p>To interpret this equation, let's try to figure out what's going on. Kaitlin is working on some number of standards. $S$ represents the number of standards she has left. We must assume that $S$ will go down as the summer proceeds. So, our basic equation would be:</p><p>$\boxed{\text{Standards left}}=\boxed{\text{Standards started with}}-\boxed{\text{Standards worked on}}$</p><p>That's just the logic of the problem. How does it match up with the equation that the problem provided?</p><p>$S=107-3x$</p><p>We know that $S$ represents "Standards left" so, so far, our equations match up.</p><p> $S=\boxed{\text{Standards started with}}-\boxed{\text{Standards worked on}}$</p><p>Now, how can we relate "Standards started with" and "Standards worked on" with $107-3x$? We know that $x$ represents the number of days that Kaitlin has worked. Each day she works, she finishes some standards. How many standards does she finish each day? Let's say that she finished 2 standards a day, and she worked for 3 days, how many standards would she finish? 6! Right. How did you know that? You multiplied them together. So, what do you think $3x$ represents?</p><p>If $x$ is number of days and it's multiplied by 3, Kaitlin must accomplish 3 standards a day.</p><p>$S=\boxed{\text{Standards started with}}-3x$</p><p>From there, you can see what 107 must represent: the number of standards that Kaitlin started with. From here, it's easy to answer the first two questions:</p><p><em>What is the meaning of 3 in this equation?</em></p><p>3 is the number of standards the Kaitlin completes each day she works. </p><p><em>What is the meaning of 107 in this equation?</em></p><p>107 is the total number of standards that Kaitlin has to write curriculum for.</p><p> </p><p>What about the last question?</p><p><em>What happens to $S$ when $x$ increases?</em></p><p>Let's think about it. $S$ is the number of standards that Kaitlin still needs to work on. $x$ is the number of days that she has worked. As she works more days, what will happen to the number of standards that she has to work on? It will decrease. </p><p> </p><p>In the first example, it's helpful to try to think, logically, about how you would try to solve the problem. Other times, it's easier to j<strong>ust translate the equation that is given, using the variable definitions that are provided in the question</strong>. </p><p><em>Let's try one more example:</em></p><p>A restaurant kitchen has to figure out prices for birthday parties. They use the expression $g(m+d)+250h$ to calculate the total price for a party. $g$ represents the number of guests who will attend the party, $h$ represents the number of hours that the party will last, $m$ represents the average cost of the main dishes that will be offered to guests, and $d$ represents the average cost of the desserts that will be offered to the guests.</p><p><em>Which of the following is the best interpretation for the meaning of the 250 in the equation?</em></p><p><em>How would you rewrite the equation if the party hosts decided not to offer dessert?</em></p><p><em>Which variable will change the most if the hosts decide to offer their guests the 3 most expensive entrees on the menu?</em></p><p>Let's start with the given equation:</p><p>$g(m+d)+250h$</p><p>Now, let's translate that into words:</p><p>$\boxed{\text{guests}}(\boxed{\text{main dish cost}}+\boxed{\text{dessert cost}})+250(\boxed{\text{hours of party}})$</p><p>So, what's going on here? If we look at the parentheses, we see that the first thing that we we do is add the cost of the main dish and the dessert. Then we multiply that by the number of guests. That makes sense: find out how much people's meals should cost, then multiply that by the number of people.</p><p>Then to that total, you add 250 times the hours of the party. What's going on there? The restaurant must charge 250 for each hour of party. So, a 1 hour party adds 250 to the cost of the food. A 2 hour party adds 500 to the cost of the food, etc. </p><p>Ok, so we understand the equation, let's answer the questions:</p><p><em>Which of the following is the best interpretation for the meaning of the 250 in the equation?</em></p><p>We already answered this: 250 is the amount the restaurant charges per hour of the party. </p><p><em>How would you rewrite the equation if the party hosts decided not to offer dessert?</em></p><p>No dessert, then you don't have to pay for dessert. Which variable represents dessert? $d$. Take that out. Or make it a zero.</p><p><em>Which variable will change the most if the hosts decide to offer their guests the 3 most expensive entrees on the menu?</em></p><p>What variable captures the price of the entrees? That's the main dish. So that's captured by $m$, which is the average of the cost of the entrees they will offer. If they offer very expensive entrees, then $m$ will be higher. </p><p>Many students find these types of word problems intimidating. Don't be afraid of them. Write them out. Label them and translate variables into things. Then, think logically: what would be added to what? What would be multiplied to what? Substitute some real numbers if that helps you.</p></div></div>
<span><span>edboost</span></span>
<span><time datetime="2024-06-19T12:45:26-07:00" title="Wednesday, June 19, 2024 - 12:45">Wed, 06/19/2024 - 12:45</time>
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<div class="node-taxonomy-container field--name-field-practice-problems field--type-entity-reference field--label-above">
<h4><i class="icon-bookmark"></i> Practice Problems:</h4>
<ul class="taxonomy-terms">
<li><article data-history-node-id="355" class="node node-type-math-practice-problems node-view-mode-default">
<h2 class="node-title">
<a href="https://edboost.org/node/355" rel="bookmark"><span>Interpret Equations</span>
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</h2>
<div class="node-content">
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>Julie is a professional gift-wrapper. She uses the expression $bp(4hl+2lw)$ to estimate her wrapping paper costs. In the expression $b$ represents the number of boxes she will wrap, $p$ represents the average cost per square foot of the wrapping paper that she uses, $h$ is the average height of her boxes in feet, $l$ is the average length of the boxes in feet, and $w$ is the average width of the box in feet. </p><ol><li>If the cost of Julie's favorite wrapper paper goes up significantly, which variable will change?</li><li>During the holidays, Julie wraps many more boxes than she does during the rest of the year. During the holidays, which variable tends to be higher?</li><li>Based on the expression, what is the best interpretation of the meaning of the number 4?</li><li>Based on the expression, which will have a bigger impact on wrapping paper costs, many boxes with larger heights or many boxes with larger widths?</li></ol><p>Garret is trying to create an expression to represent the number of pages of summer reading he has left. He has come up with $L=3670-64d$ in which $L$ represents the number of pages that Garret has left to read and $d$ represents the days of summer that have passed.</p><ol start="5"><li>In the expression, what will happen to variable $d$ as the summer progresses?</li><li>As $d$ increases, what will happen to $L$?</li><li>What is the best interpretation for the meaning of 3670 in the equation?</li><li>What is the best interpretation for the meaning of 64 in the equation?</li></ol><p>The city lake rents boats to tourists in the summer. A guide book tells visitors that they can use the expression $C=8+5h+3r$ to calculate the cost of their boat rental. The guide explains that $C$ represents the cost of the boat rental, $h$ represents the hours of the rental, and $r$ represents the number of riders in the boat.</p><ol start="9"><li>In the expression, what is the best interpretation for the meaning of 5 in the equation?</li><li>What is the best interpretation for the meaning of 8 in the expression?</li><li>Who will pay more for the boat rental, a group of 8 who rent the boat for 2 hours, or a group of 2 who rents the boat for 8 hours?</li><li>Would it be cheaper for a group of 8 to rent one boat for 4 hours or two boats (each with 4 people) for 4 hours?</li></ol></div></div>
<div class="node-taxonomy-container field--name-field-answer-key field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Answer Key:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/node/134" hreflang="en">Interpret Equations</a></li>
</ul>
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</article>
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</ul>
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<div class="node-taxonomy-container field--name-field-skill field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Skill:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/taxonomy/term/57" hreflang="en">Word Problems (Algebra)</a></li>
</ul>
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<div class="node-taxonomy-container field--name-field-common-core-grade-level-su field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Common Core Grade Level/Subject</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/taxonomy/term/16" hreflang="en">Algebra I</a></li>
</ul>
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<div class="node-taxonomy-container field--name-field-edboost-test field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> EdBoost Test:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/taxonomy/term/4" hreflang="en">Algebra I</a></li>
</ul>
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Wed, 19 Jun 2024 19:45:26 +0000edboost581 at https://edboost.org