-
Notifications
You must be signed in to change notification settings - Fork 0
/
12702f20-bb71-4741-9a07-3175a8c2e3d5.html
178 lines (178 loc) · 8.97 KB
/
12702f20-bb71-4741-9a07-3175a8c2e3d5.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
<h4>Activity (15 minutes)</h4>
<p>In the previous activity, students recalled what it means for a number to be a solution to an equation in one
variable. In this activity, they review the meaning of a solution to an equation in two variables. </p>
<h4>Launch</h4>
<p>Arrange students in groups of two to three. Without giving away the equation, present the scenario and ask students
whether they are familiar with this kind of situation. Then encourage students to use Desmos to model the scenario
with an equation.</p>
<p>As students work, quietly ask one or two students if they would be willing to share their reasoning with the class.
After they share out, discuss why the values in problems 1 and 3 do not satisfy the equation (or appear on the graph).
</p>
<h4>Student Activity</h4>
<p>Use the following scenario for questions 1-3: </p>
<p>One gram of protein contains 4 calories. One gram of fat contains 9 calories. A snack has 60 calories from \( p \)
grams of protein and \( f \) grams of fat.</p>
<ol class="os-raise-noindent">
<li>
Determine if 5 grams of protein and 2 grams of fat could be the number of grams of protein and fat in the snack.
Explain your reasoning.
</li>
</ol>
<p><strong>Answer:</strong> No, because \(4(5) + 9(2) = 20 + 18 = 38\), not 60.
</p>
<ol class="os-raise-noindent" start="2">
<li>Determine if 10.5 grams of protein and 2 grams of fat could be the number of grams of protein and fat in the
snack. Explain your reasoning.
</li>
</ol>
<p><strong>Answer:</strong> Yes, because \(4(10.5) + 9(2) = 42 + 18 = 60\).</p>
<ol class="os-raise-noindent" start="3">
<li>Determine if 8 grams of protein and 4 grams of fat could be the number of grams of protein and fat in the snack.
Explain your reasoning.
</li>
</ol>
<p><strong>Answer:</strong> No, because \(4(8) + 9(4) = 32 + 36 = 68\), not 60.
</p>
<ol class="os-raise-noindent" start="4">
<li>If there are 6 grams of fat in the snack, how many grams of protein are there? Be prepared to show your
reasoning.
</li>
</ol>
<p><strong>Answer:</strong> 1.5 grams. For example: \(4p+9(6)=60\), so \(p\) must be 1.5 for the equation to be
true.</p>
<ol class="os-raise-noindent" start="5">
<li>In this situation, what does a solution to the equation \(4p+9f=60\) tell us? Give an example of a solution.
</li>
</ol>
<p><strong>Answer:</strong> It means a pair of grams of protein and fat in the snack that add up to 60 calories.
One example would be 6 grams of protein and 4 grams of fat.</p>
<h4>Video: Working Through the Equation</h4>
<p>Watch the following video to learn more about how to determine a solution to this particular equation: \( 4p+
9f = 60\).</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/ce9b7554a5f6c2231ff7bae0a6c1e09f8ad55e39">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/92159e35ce10b76ecda963022a98325d7722afac " srclang="en_us">https://k12.openstax.org/contents/raise/resources/ce9b7554a5f6c2231ff7bae0a6c1e09f8ad55e39
</video></div>
</div>
<h4>Activity Synthesis</h4>
<p>The goal of the discussion is to make sure students understand that a solution to an equation in two variables
is any pair of values that, when substituted into the equation and evaluated, make the equation true. Discuss
questions such as:</p>
<ul>
<li>“In this situation, what does it mean when we say that \( p=12 \) and \( f=1.5 \) are <em>not</em>
solutions to the equation?” (They are not a combination of protein and fat that would produce 60
calories. Substituting them for the variables in the equation leads to a false equation of \( 61.5=60 \).)
</li>
<li>“How did you find out the grams of protein in the snack given that there are 6 grams of fat?”
(Substitute 6 for \( f \) and solve the equation.)</li>
<li>“Can you find another combination that is a solution?”</li>
<li>“How many possible combinations of grams of protein and fat (or \( p \) and \( f \)) would add up to
60 calories?” (many solutions) </li>
</ul>
<p>As a segue to the next lesson, solicit some ideas on how we know that there are many solutions to the equation.
If no one mentions using a graph, bring it up and tell students that they will explore the graphs of
two-variable equations next.</p>
<h3>1.4.3: Self Check <br></h3>
<p class="os-raise-text-bold"><em>Following the activity, students will answer the following question to check
their understanding of the concepts explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p> Which of the following is a solution \((x, y)\) to \( 2x + 3y = 7\)?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>\((2, 1)\)</td>
<td>That’s correct! Check yourself: When \( x=2 \) and \( y = 1 \), \( 2(2) + 3(1) = 7 \) is a true
statement.</td>
</tr>
<tr>
<td>\((1, 2)\)</td>
<td>Incorrect. Let’s try again a different way: Let \( x = 2 \) and \( y = 1 \). The correct answer is
\((2, 1)\).</td>
</tr>
<tr>
<td>\((4, -1)\)<br></td>
<td>Incorrect.
Let’s try again a different way: When \( x = 4 \) and \( y = -1 \), the equation becomes \( 2(4) +
3(-1) = 7 \) or \( 5=7 \), which is not a true statement. The correct answer is \((2, 1)\).</td>
</tr>
<tr>
<td>\((-2, 4)\)<br></td>
<td>Incorrect. Let’s
try again a different way: When \( x = -2 \) and \( y = 4 \), the equation becomes \( 2(-2 )+ 3(4) = 7 \)
or \( 8 = 7 \), which is not a true statement. The correct answer is \((2, 1)\).</td>
</tr>
</tbody>
</table>
<br>
<h3>1.4.3: Additional Resources</h3>
<p class="os-raise-text-bold"><em>The following content is available to students who would like more support based
on their experience with the self check. Students will not automatically have access to this content, so you
may wish to share it with those who could benefit from it.</em></p>
<h4>Solutions to Equations in Two Variables</h4>
<p>An equation that contains two unknown quantities or two quantities that vary is called an equation in two
variables.
</p>
<img alt="Drawing of equations of two variables. f equals nine fifths c plus 32 to convert temperature from degrees fahrenheit F to degrees celsisus C and a formula i equals two point five four c and coverts a measure from cenimenters c to inches i." src="https://k12.openstax.org/contents/raise/resources/331e9b9ec3b58b0994b05924f6aec109de692a5f" width="400">
<br>
<p>A solution to such an equation is a pair of numbers that makes the equation true.
</p>
<p><strong>Example</strong><br>
Suppose Tyler spends $45 on T-shirts and socks. A T-shirt costs $10, and a pair of socks costs $2.50. If \(t\)
represents the number of T-shirts and \(p\) represents the number of pairs of socks that Tyler buys, we can
represent this situation with the equation:
</p>
<p>\(10t + 2.50p = 45\)
</p>
<p>This is an equation in two variables. More than one pair of values for \(t\) and \(p\) make the equation true.
<br>
Which pair of values makes the equation \(10t + 2.50p = 45\) true?
</p>
<ol class="os-raise-noindent">
<li>\(t = 3\) and \(p = 6\) </li>
</ol>
<p><strong>Answer:</strong> These values make the equation true, because:</p>
<p>\(10(3) + 2.50(6) = 45\)<br>
\(30 + 15 = 45\)<br>
\(45 = 45\)
</p>
<ol class="os-raise-noindent" start="2">
<li>\(t = 4\) and \(p = 2\)
</li>
</ol>
<p><strong>Answer:</strong> These values make the equation true, because:</p>
<p>\(10(4) + 2.50(2) = 45\)<br>
\(40 + 5 = 45\)<br>
\(45 = 45\)
</p>
<ol class="os-raise-noindent" start="3">
<li>\(t = 1\) and \(p = 10\)
</li>
</ol>
<p><strong>Answer:</strong> These values make the equation false, because:</p>
<p>\(10(1) + 2.50(10) = 45\)<br>
\(10 + 25 = 45\)<br>
\(35 \neq 45\)
</p>
<p>In this situation, one <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="storename">constraint</span> is that the combined cost of shirts and socks must equal $45.
</p>
<p><span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="storename">Solutions</span> to the
equations are pairs of \(t\) and \(p\) values that satisfy this constraint, such as in questions 1 and 2.
</p>
<p>Combinations such as \(t = 1\) and \(p = 10\), as in question 3, are not solutions because they don't meet the
constraint. When these pairs of values are substituted into the equation, they result in statements that are
false.
</p>
<h3>Try It: Solutions to Equations in Two Variables</h3>
<p>Is \( a = 3 \) and \( b = 5 \) a solution to \( 6a -3b = 3 \)?</p>
<p><strong>Answer:</strong> These values make the equation true. They are a solution to the equation, because:</p>
<p>\(6(3) - 3(5) = 3\)<br>
\(18 - 15 = 3\)<br>
\(3 = 3\)
</p>