Subtraction in Equivalent Expressions: IM 7.6.18
[url=https://im.kendallhunt.com/MS/teachers/2/6/18/preparation.html]“Subtraction in Equivalent Expressions”[/url] from IM Grade 7 by [url=https://www.geogebra.org/m/gus8ffvr]Open Up Resources[/url] and Illustrative Mathematics. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0 license[/url].
Table of Contents
- Lesson 7.6.18
- IM 7.6.18 Lesson: Subtraction in Equivalent Expressions
- Practice 7.6.18
- IM 7.6.18 Practice: Subtraction in Equivalent Expressions
IM 7.6.18 Lesson: Subtraction in Equivalent Expressions
Find each sum or difference mentally.
[math]\text{-}30+\text{-}10[/math]
[math]\text{-}10+\text{-}30[/math]
[math]\text{-}30-10[/math]
[math]10-\text{-}30[/math]
Lin and Kiran are trying to calculate [math]7\frac{3}{4}+3\frac{5}{6}-1\frac{3}{4}[/math]. Here is their conversation:[br][list][*][b]Lin[/b]: “I plan to first add [math]7\frac{3}{4}[/math] and [math]3\frac{5}{6}[/math], so I will have to start by finding equivalent fractions with a common denominator.”[br][br][/*][*][b]Kiran[/b]: “It would be a lot easier if we could start by working with the [math]1\frac{3}{4}[/math] and [math]7\frac{3}{4}[/math]. Can we rewrite it like [math]7\frac{3}{4}+1\frac{3}{4}-3\frac{5}{6}[/math]?”[/*][/list][br][list][*][b]Lin[/b]: “You can’t switch the order of numbers in a subtraction problem like you can with addition; [math]2-3[/math] is not equal to [math]3-2[/math].”[br][br][/*][*][b]Kiran[/b]: “That’s true, but do you remember what we learned about rewriting subtraction expressions using addition? [math]2-3[/math] is equal to [math]2+\left(-3\right)[/math].”[br] [/*][/list][br]Write an expression that is equivalent to [math]7\frac{3}{4}+3\frac{5}{6}-1\frac{3}{4}[/math] that uses addition instead of subtraction.
If you wrote the [b]terms [/b]of your new expression in a different order, would it still be equivalent? [i]Explain [/i]your reasoning.
Write two expressions for the area of the big rectangle.[br][br][img]data:image/png;base64,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[/img][br]
Use the distributive property to write an expression that is equivalent to [math]\frac{1}{2}\left(8y+\left(-x\right)+\left(-12\right)\right)[/math]. The boxes can help you organize your work.[br][br][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAV4AAABoCAYAAAC5f5geAAAfxUlEQVR4Ae1dCXsUxda+f+QjO0lIwpYEwr7viCCigCKIO9x7lassKuDOTljdAHEBQRZFrwoKyBLWiyC7IEKAQNiXsC9J5nzPe2aq09PTk5me6Z4e4Mzz9FR1ddWp6vdUvV1ddar6HyS/2CLgMc/O4wlywTy6hAoCgkAQBNCS1BEkiuvB/3C9BPdRAYKSp45TVZx4rxj3kVrkVuIcAdUmoi2mXXKiLUc46YV4w0Epyjg63mVJ91IFifLWJbkgUC0CxrahInN4sIsqkolrbFtKhBZeqUJMEscwSIjXAbChZBwVFRV0p/wu3b17l+7cucPn5FO8Xv1apXCgLCJSELjXEEB7wVFeWRFW0Y3tB+dIW37H2+7Q9lQc1e7UeVgZOBBJiNdmUCsrK+nGjRtUWlpKf/zxB20s2sDHpg0badeuXXT2/Dm+rgjY5uxFnCBwzyIAMrx25Spt3byFijZuoMN/HbJ0L+Xl5XS17AqdLDlBO3f8QRs2VLW9P//8ky5fvkx3bt22JNOpyEK8NiKLJ+uJEydo1ief0sABT1N+/VyqmZJKNVPTKD01jZo2akyDXniRFsybTyeOlxBIWv9z+ymsL4v4BYFYIgDSPH2ylMaPH0+ZNdMpNSmZhr02NKwiIG3ZpctM2Ejfr18/bnuZqTW5/aUlp1C71m1Y3ooVK+jq1athyXUykhCvTeiCRLdv306PPtKTUhKTqHZWNrVq3oL6PfEkPfVkP3qiT19q0aw51c7MouSEROrzeG86cOBAQO5CvgGQSMB9jgB6oXgb7N/vKSbc1IQEbkNvjHg9rDv/68BB+s8rQyg3N5dJO7duPerUoSO3O7S/J/v0pYL8Bnwtu1YWvffOu1RWVhaWbKciCfHahOy5c+eoZ49HuOLUzalNE8aO49cdJR5P5T+276DhQ4dRrfQMjvfCs895hx1UJHEFgQcMAQwt/LJ8BXVo244yk1KpUcMCqlu7jpd4h48ICw28YebUyiK0O7SppYuX0LHio1patL21v63ht1C8eaYkJdNHMz/0zrlosWLrEeK1Ce8ff/gvKzQtMZkmTZjolWoyg4oJt8GDB3PFSk+rSUXr1odVAq0nrGYHkErvD0uKRLKEgA9fDXtfYgSrw5I8ieyHwNmzZ2ncuHFMmhheeKhzF/rpvz9ybxVDDa+HSbwYux06dCjNnTuXrl+/7peH/gT5dX+oG7e9xgWN6Pjx4/rLMfUL8doANxrm5ImTuBeLIYa//vorQKqeI/GEx3BEcmISzZoz2y8uns7q8LtgcoJ8PRWVVVYTnkpt9tYkugSFQEARLMbqNSsUfZpKj5dwdYtdlMUKdCa/8BAAzhiamzhxIreZ7Mxa3BvFkMH58+epTavWhA5MuMSLXG/fvh1W3f/y8y8oLTGV813564rwCuxALCFeO0D1EE8KpCanUF79XDpy5Ei1Urdt28bjvCDfD6fP0OJWllfQ/v37aePGjbTj9+3c+NVFRQrqHC7af/Gxoxx/4/oiunzhIl9mQtaRgz6N+A0IGHq1INLt237nGfGdO3dWOwFaeuIkW6xg9hx6UzpSriEnOeU6W9UF+emnn3je44P33qcrV64wfidPnqSWzZozMVohXoAbDu5FRUWEMeTUxFT6+MNPXNOJEK9N0C9a+A1h9jQzPYNWrVqlVQJjZcBQw+xPZ3GPNysjk1auXKmVAJMMLzz3PM/EtmjclM6cOaNdM/NcvnyRewp4LcvPzfNr/GbxJSw0Ardv3uJJHjwUWzZvwdYnZqkuX7xEg18axHrEhM3cOZ+ZRZMwHQL6tgA/2sKWjZvYxAvnoGQQLyalrQw1IAu9bF2WAd61q3+jmjUzKCUxgaYWTgm4HqsAIV47kPYQHSs5ziYrSQmJ9FjPR+no0aM8BADxlb7BWPSmYMnQsV17JtcnHuvNdouqCHj9GvvBGG7MOTk59Hc1doyoaIsWLuYKCnO18WPGBlS+qr6FykHcUAgAs9W/ruQZcLzuLlqwkPQ218D91q1bNHH8BEpNSOKJUvj1cULlIderENATJvywf2/dzEu8I4YNr4oYxGdMHySaN9hDtGD+19zjTUlIoQULFlQb3cmLQrx2oeshmjdvHsGUBbOmvR97nNas/o0uXLjA40/oIWERBSwfYNXQvm07Hk7QKg4m4lTFSEpm8t26datWOi2eLwQTCo/26M7xYKOIngJ+Kp5yNQHiCYmAekDC1Khn9x6Mbd/He9OtGze1tBjL/W7pt5STlc322ej1Xrt2TbsuHosIGHoHJSdPcI8XbxxWhxqC5uxtWnTz1h365wsvsV6zMtJpx44dQZM4fUGI10aEMUb79bz5PEaFXmjt7Bw2b/n5x5/o1SH/YTMZTL49O/AZ2rdvn5ckUSl047GrV67iXixetb5Bb8vkh/jzvvyKhzYwvDFnzhwtll6WFiiesBFg/DxEX33xJeObUTOdh44gAG8kO7b9Tk0KGvG1J3r3odOnT4ct+0GOGG69VEMN4RKvkqvc6jDevHkz1apVi99Uej7cky5dulRddEevCfHaDC9mw7///nsmX1QeNFy2HUxM4iftyy+/zGYsaMS+EQi/Euzdu5fq1anLcadMLvS7pk6O/H2YhzUgHz1o9KrlZxMCvjePSxcuUtdOnfkhOOjFlwhjv3/u208Pd32Iw3r1fJQOHjxoU6YiRiGAoQarY7zhkC7sevEWijaDzs+yZctUlq64Qrw2wo5X0k2bNvEqGhhzF+QVcO/2yb5PUOOGBUzALZo3pTHvf8A9XpC08YceVLPGTdjq4bX/vGq8zKZm48eO46c2zHAWL14cEEcCQiNg1lj9wjzEk6BY8t0wL58nQUHAaLitWrSkbVv/FzqTByAGRgoMowV+d+2Hqd8V8xPV47U8uWYijvP2EJ0/e47efP0NfkuBPl/598s8/GeSJGZBQrxRQs0Vz+OhmzdvEoYUMHEGY/CHH+pGn348i19F8bSF/SCWRDbIy2dS7f9kP9qyaTMTqSoCKgrGELt27sKzrujNcs9Yi0B06NAhJnHYAD/79EDe+ENdhmu1ouvT3q9+6EZ/YBMj/bn+vvX4HT58mJo1acpk27lDR54Nb9K4gJYskoedHjP4YaEATG/cusmrMeHqMTbrZBhl4DxS4kVave5YtocIby5TJxfyvAoemlhAgQ1z3P4J8dqkAexChp4qhhWwD8PunbtYspqwAUGfOXOOZk6fwcSJJ3rXTh3ZBExfBFSe5597hhs7GjnsG9UPr7sTxo3nV10sq1y7dq26JG4QBGCBgMmwb5csDXARhmEh9XAz9twwZv/WqNGazTWWpWLsV8UPkuUDGXzq1CkN42VLvyUcCvelS5fS+vXhrdCMhHiNelMKwCQplhPDxBMWKG1atiLY8fLPN6Sk4sbaFeK1AXEsU+TX0KRkNgjHngxV71+Ker0ZwaTsk48+pto5tSmlRiINf22od6s63SQbTMpAzHn16hNW86iKhW0muQeWlEz/HvxPHnfUFx+krX/q6/36eA+SHw8uNDxMQvJOcSmp7FfnsEKBTsywAmkXFhayLrCxUbcuXV3fXCUudechXkiCCWXgajyA+zNPD9S1CZPeqe/GIiFeJFVthP0eD5WVXeK3zNy6dZh0mzZuwg+CeMFPiNcGTezfs5cy0mpyL1XtqORtyP7bPqqs8KqLighyRYU4dcZ/ZhyWEWjotTIyad26dUwKeFWDZQSe3A1y82jLli1KnLjVIICHIsbYYVv9+KO9CJNij/lc7CT33MBnvMSrs7eGOLw6F61dx4soMKwDXTVp1Nh0OXg12T8wl3b9sZPxBbb6A5jjwLxEOLbOkRKvHmjo/Ksv5lGd7ByCvS70Bgshns/2+DZXN9lHRS/Dab8Qb5QIg2CXffsdk25aclKACZj2JFYen/vxhx/xskUQ9t7de6pK4SHeOAfEi2GLH5Z9z9ewsTN6ZyCBUW+O1BZnVCUUnxkC0A/G9GCRoD/2793H5/xGocz5lI6IaM+ePbzFJ3prGK/HBBvGCKdPnSZDDSZAg+yAGTAGtgpfnCM83A1poiVebIS+aNEialLQkPXVvElj3v/a7I3G5DZiFiTEawPUmGwBUaKRYoLN+PMfbPBenf/VPO5F4bVsz7693kBfwy8+fIRqpXt70F9/NY8n3Ea+8SbHx7Z5mGCTX3gIhGpwgNwYByZNT/cfwA23XZu29L8tW2nIy68w/m1bt6HTpafCy/wBiWWGYaS3Hg3xXrt2g7B0H2+EeEiip7twIXq65m+ekZbRjnRCvFGiiEaLibWaKcmUmpRIM6ZNDykRY4qjR47ihowNmlVvQBHAhQvnfKt3EnhyAMuPGzVoyMQ+9v0PqiwhdD00faZKjj7sQfeHiwn2VX737XcYa1igYIIIvzWrVhP21sDDdc6s2XHZmO8HHUdKvOjpLlmyhBe3gHTxtRcM2Rn1bjx3CzMhXhuQx2Y2MPrGUuFePXqxCZlSsJEbMXaI8bAWTZsx8WKsF69p/PNFxqTOgKf68/VJ4yfQtClT+QmOWdlDh/+2ocQiwgwB6AFDCRj+gQUDNlGBvvDDuPyTfXqzTrp3e5jNlMxkSFh0CJwotb47GZZsL/5mETXMy2X9YN4EY7pKd1ZKpNqtlTSRxBXijQQ1QxooeNrUQsrKrMU937dHv0V7d+9j8zFly3jj2nWCyQ0M72F/i6cyerFqDFcTWem15cX3ptC7ev6ZZwk2pCAD7JovP/sRwFAQ9nOFaVn9uvV4teG///kvunjRu82myhGNGysRsdfGihW/qmBxbUQAPV7sCofJzHD2aoDe8FaixnRBvp/NnuNnQ6zaoJmLt083fkK8NqGOFWdYEQMTMOxq1f2hrjTmvXfZthH2omi0GF7AmCHPkDdoQB/OmOk/SabrHsPeF/Hq16nLDR3fkMIKHPnZjwAsRtavXae9tWBW/tixY1pGSi3QMRbI4KH5zID+vEuZFkk8tiCAj8XyfrwJSfyZrFBCMQwHooZOcEB3WM2JNhfOsWfX7lBZOHJdiNcWWL1ffsDG2FjggPX8LZu2pHq16/C4YHZ6JmF5L0gZlQRrxufMmVv1vTXVsnVlwVpyTLxh0g4r4fQb4eiiidcGBLD9Zt/efVhXrVu0pHVr1nrHBv304p0inVY4hdf659bLI2xoLz97EQDx4vtr2P1t1BtvhhQOqwksx8f4u/6AKab+PJh/woQJIfNwIoIQr82oYgYVvaUffviBxo0ZS0NffY2f3HALJ02mlb/8SiUlJWw1iqz1bVuNL8HFFyXweWr0env3ekwM923WkxIHfWGIAUM7rw8bTvh2XnW/40ePsU6HDX2V8AUF+UWHgKrzkAI/tjtF52XEq0N5siyU9FMnS+md0W+xTka8NoL1CF1iYRLcUMfyn372y0K1R325/CLYdCLEawOQeiUpxSmx+msI8zs3RvYlwpgxzM3Q08UkT8A4sBIubtQIKBVAL3668Un2KIN7lZNvb1ecmsVX0cSNDAG7MLVLTmR3ETqVEG9ojOyLoVp5MIm+6ydLTmhfQ4U9KW+0rdIqN5gMCbeMQKhGqoc8VFzLmUsCDQEjtnrctUj3iUeIN84UCVMyZT6GlWorfl7OJTRWyjgr9n1ZnADM1X4a9zMjuK3JCLEN0JXb9xEifyHeEADF8jLGG7EHA1anwZQM41N6c5d7rXLFEjvJSxC4lxAQ4nVJWyBRfDkCtoUwzsekwvbt27Uv3Hbp1Jk/jKknW73fpWJLtoKAIGADAkK8NoAYrggjcWICbeniJWxviE++w0YUNsBYBYc9HyJ86wq3OBJPEBAEXEJAiNcl4JHtkCFD2L4XNr6wM2zetBn1eqQnr8QBSSuiVq6LRZWsBQFBwEYEhHhtBDOUKGMP9rdVqzU7w+FDh9E3Xy+g4uJiTYwxvnZBPIKAIHBPIyDEG6fqk15unCpGiiUI2ICAEK8NIFoRYUao0rO1gqDEFQTufQSEeONEh6bkaxoYJwWWYggCgkDECNzzxGvWg4wYDUkoCAgCgkAMELjniTcGGEkWgoAgIAjYhgA6i0K8tsEpggQBQUAQMEdAvZkrV4jXHCcJFQQEAUEgagQU0UKQ8sO954lXfzNRoyQCBAFBQBBwEAHFV/c88SqM1A2pc3EFAUHAHwFpI/54uHl2zxOvVCY3q09g3qKPQEzcDBF9uIl+8LzveeLFtolXr17VjitXrmh+fbj4qzByEgvsslZWVsY6wAbuTuYlskPrFPqQNhEaJ6fqErDXtwN8WBW/0MRrYsRvEhSc2h28cuHceVq6aDFNLZxCM6ZNp+lTp/GBjcSVX1wvJs7hMEPDesrkQvrgvfepcOIkX1jVNefyd/r+7g35ZnV+8oTxrA/jtapz0U+s6qXCHF8+BhmHJl4HiTNa0dg6sV3LllQnO4vq1c7ho25OtuZXYeJ6sXEKh/p1avMXlevm1Kb0lHTecc2pvESuvy7r5/if6/HBrnfpqWlUJzuHoBuvnoLH16cVv304VbWPbGrTshX9snxFeMTrN06k7+7q/dGyaATp0dNNTUjiT6CjYjVq0FA78MlnHPow8VfhYxcWeoxrpiSzLpJrJPCn7JsUNBL8dXXSLsz1cpo0qKrjyg+d5NfP1XSBr5k0zMunxvkN/dqEXnd6meK3r50U5DcgtAN8xis5IZEPvBWG3eP1I1+QpMukiyLg1RafP09JTKIhL79C8778ir78/Av66osvtQPncjiHAWPtwzi/Xj6lJCYQCHj0G6Pp63nzBfsY1D9jfUc7mDh+EjdytA00+o9mfshfrpa24FxbMMMWukE7GDFsuKaPkW+8GT7x6om2tLSUdu7cSaUlJyLop9qXBD1eVKykhERa+PUCwsA1JtvkiC0G5XfuMvatmjfjB2FachKtW7Ne9BGjuqiv9/Dj2LN7h9bQ8UWTs2fP0u3bt7lt6ONLW3G+rQBvDIuqHu/okaMsEC/4stJDe/fupVFvjqSB/QfQF3M/t49FI5BUOGkyN3TcED6hE9Arj0CmJLGGgMK8kjzUpnlzfhCmJ6dSUdFG/bPamlCJHRECShdIfGD/QUqu4X0bbN2iJV28eDEimZLIHgR+XfFLZMSLL+AWrVtPzz/7HGVl1mLCA3O7+VM9XhDvkkWLA4qir4gBFyXANgSAM+pH6xbNKTmhBqUlp1BRUZFt8kVQeAh4PBVaxD/37edhH7wRYkIHH1aVX+wRUByECTVLPV4kxNNy2bffUe+evfi7YE898ST3bN4e/Vbs78SXI8qlEW+NBO7xulYYyZgqK4mJFw0dkzlCvO5WChAvGjr0gR6vnngVGbhbwgckd99cWETEO3/+fGrXpi21bd2GJ7BmffIpK/Sdt952FT1MrqFiBevxulq4Byxz1eMV4o0PxVdHvPFRwgerFBER75xZs6lv7z70/XfL6Pr160y+sCYQ4n2wKk91dyvEWx06sb8mxBt7zKvL0TLxQtjJkydp6+YtVFHhHUOC2YQQb3UwP3jXhHjjS+dCvPGlj4iIV92CGhsS4lWIiKsQEOJVSMSHK8QbH3pQpYiKeJUQmJFJj1ehIS4QEOKNr3ogxBtf+rCFeKXHG19KjYfSCPHGgxaqyiDEW4VFPPiiJl4MNwjxxoMq46sMQrzxpQ8h3vjSR9TEi9sR4o0vpcZDaYR440ELVWUQ4q3CIh58QrzxoIX7sAxCvPGlVCHe+NKHbcQLQ3mx440v5bpZGiFeN9EPzFuINxATN0MiIl5lRqZcsWpwU4XxmbcQb3zpRYg3vvQREfEab0HGeI2IyLkQb3zVASHe+NKHEG986eO+KY0Qb3ypUog3vvRhC/Fi7wbsQIWd1N38ySY5bqLvn7cQrz8ebp8J8bqtAf/8oyJeNca79rc19K9Bg2nRwm/8pcf4TIg3xoBXk50QbzXguHBJiNcF0KvJMiriVXKvll2hkqPH6OJ5dzdXFuJVGnHfFeJ1Xwf6Egjx6tFw328L8bp/G94SCPHGiyZkr4b40YS3JEK88aWRAOJds/o3/lovvuYwbcpU/qqD8uvPEaY/jNdAgvrrej/iTp44iTZu3Gjrd9GEeOOnckmPN350gZII8caXPgKId+wHYyivfm7oIy+P4+Tnel2VJjc3dNrcevWpQW4eTZ1caCsaQry2whmVMCHeqOCzPbEQr+2QRiUwgHjff+ddys6sRTm1stg189fOzKKcDG+crIzMgPhIo9LhQ5h6WThX16dOnBxV4Y2JhXiNiLh3LsTrHvZmOQvxmqHiXlgA8W7ZtJnmzvmMPv9srnYYz9U1Fa5chGMVG1wV9gXCdLI47ZzP6LPZc2jb9t/5zpV1RLTf/xbida8iGXMW4jUi4u65EK+7+BtzDyDemzdv0pUrV6isrMz0UNcuX75seh0WDsa0xjQ4v3TpEt26dctYnqjOhXijgs/WxEK8tsIZtTAh3qghtFVAAPGGKx0N68TxElq9chWt+Hk5H7t37qJr1675i/B9ztgv0CzML0JkJ0K8keHmRCohXidQjVymEG/k2DmRMmzi1Q8H7N29h8a8/wH1fuxx/sR721atCUfnDh1p0KBBtH79+mqtFYLxbrDwcG9ciDdcpJyPJ8TrPMZWchDitYKW83HDJl5VlO3bfqee3XtQ/Tp1qVnjJjSw/wAa/NIgeuG556lRwwLKrJlOnTp0JMSL9U+IN9aIB89PiDc4Nm5cEeJ1A/XgeVomXiwPbtKoMb01ajStW7eODh8+TMePHqOjR4pp7dq11L/fU/zhy0EvvkS3b9/25hxtVzZ4+f2uCPH6weHqiRCvq/AHZC7EGwCJqwGWiffChQu0YX0RnS49RXfv3uXCq2GIO3fu0Jo1a9hcrHFBIyopKfG77vSdCvE6jXD48oV4w8cqFjGFeGOBcvh5WCZekKwiXLNsSktLqUlBI6pXuw79deCgWRTHwoR4HYPWsmAhXsuQOZpAiNdReC0Lj4h4jbmoHi/ckyUnKLduPV6Zdqz4qDGqo+dCvI7Ca0m4EK8luByPLMTrOMSWMrBMvKGkL1++nPfmHfBU/6ox3lCJbLouxGsTkDaIEeK1AUQbRQjx2gimDaIsE6/q3SJvvR/nWDgxcMDTPMb73+9/oGhXolm9PyFeq4g5F1+I1zlsI5EsxBsJas6lsUy8ZkWB0UJFRQVNK5xCtbOy2eLh2pWrZlEdDRPidRReS8KFeC3B5XhkIV7HIbaUQVTEq6zEPOUVtHTxEmqQl099Hu9NsR7bVXcsxKuQcN8V4nVfB/oSCPHq0XDfH5J4jcMJimwrfeMIaGAzp8/gCTXY7h4/ftxvCMKY3slbFuJ1El1rsoV4reHldGwhXqcRtiY/JPGaivOxb2V5Be9EVic7h/r160fHjh0jNDj1U6SrXBXulCvE6xSy1uUK8VrHzMkUQrxOomtddmTES0R3ym/TzGnTeenwsNeGshmZnnRVUWJFushPiFeh7r4rxOu+DvQlEOLVo+G+PyTxmhEnVqjNnDqT6tbJoZdeeJFOnz5NFZ7Anm6sb0+IN9aIB89PiDc4Nm5cEeJ1A/XgeYYkXmPS8vJyGjdmLDXMy6Xx4yfSxYsXyYOhBzX4a0hgRtyGKLadCvHaBmXUgoR4o4bQVgFCvLbCGbUwS8QLk7HpU6dRdq0sKsgroNmfzqIlixYHPbChTnXLi6MuvUGAEK8BEBdPhXhdBN8kayFeE1BcDLJEvFevXqWCBg1597GMlDTKr1uf8vFhzHr1tQNhOEf4oz0eoetXDRujO3izQrwOgmtRtBCvRcAcji7E6zDAFsVbIt4bN27wBujDhw4j4zFi2HC/MJx/OGOm9/M+QYYhLJY1ZHQ98X67ZKkWP5bDHVqm4qHWLVpSSmISLyHftGlT0OEogcpmBNTQn67dHfzzACUnJFJKQgq1admKyi5dtjlTERcWAj7drPzlV0qpkcjHWyNH0T/0iY2EhXPj99OM31fTf4sNnwEyytDLt9uPz8WjoSclJNKcT2fRmTNneOJPuafPnuEwnMvhLAanTp3ijfJTk5IpLTmFfvzxR8HchXqn6vzGog1e4k1MohZNm9Ghg39pbYPbx9mqtiJtw9m2cfbsWVq08BvmKjwMRxuJNxxiBLGakasxzHgejuyQcXRPdMSdPHGS92b+L4ErF4Y6euLo3oMe6d6DXfjlcA6DHg931/DOyMiglMQESq7xf9S+TVseehLsncPer573eESr86j7ndp3YOJNTUigrIx06talq187QBxpI87qRumnV89HqU2r1tqDcNSbI/17vCGJTxfBEWLVyTd6zfLDxB96WClJyezWTE3j19y0lFQ/t6bvXFwvLnbioLBOT02j1IQkfhDCtTMPkVW93qADpQfNTU5hXeCNMLmG9y0E16An4JmWmiw6iiUvJHn1gR5v4cRJ1onXSIDa8FKQnrCRQK2e++Vn6PHu3r2bnujbm7p07EBdu3Sihzp2pK6dOlLXDh3Y1c4RJoejGHTr1IXat29PjfPzqX2b1tStUydH8xN9qjrdmR7qqI6OvjbQmdtDm9YtqVGDhtSpY3vu8UJHwM3bLjqLfmLICV06deZvU/bt3YcOHTpURbx+BGdgx2DXgoUbkjtyirzL79zlTxIdOXKEjvx9mIqLiwl+5XI4rsnhIAZ/s+yjR4+yDvbs2kuHDv7t1cFRrz4E/9jVwcPF3rzwTcSDBw/S7p27tDZRXHxY84tOYqcT8FHxYS9H8eKzu+VVxGsHOyoiVq4dMv1kVBq6vL6LngrvKjrkK0dsMYAKFOZYzQjbb5iWqTBxXdRHRYWfPrDoCfrgX2Vsy/Vg1wMv7mgX3F6I7CFeTZleldr6b5StPw/q95VAf93WQokwPwT8ca5aSu4XSU5ihoC/PoJkG6QTEyS2BNuIAPTz/2QsTJwVFsamAAAAAElFTkSuQmCC[/img][br]
Use the distributive property to write an expression that is equivalent to [math]\frac{1}{2}\left(8y-x-12\right)[/math].
Here is a calendar for April 2017. Let's choose a date: the 10th. Look at the numbers above, below, and to either side of the 10th: 3, 17, 9, 11.
Average these four numbers. What do you notice?
Choose a different date that is in a location where it has a date above, below, and to either side. Average these four numbers. What do you notice?
Explain why the same thing will happen for any date in a location where it has a date above, below, and to either side.
IM 7.6.18 Practice: Subtraction in Equivalent Expressions
For each expression, write an equivalent expression that uses only addition.
[math]20-9+8-7[/math]
[math]4x-7y-5z+6[/math]
[math]-3x-8y-4-\frac{8}{7}z[/math]
Use the distributive property to write an expression that is equivalent to the expression. If you get stuck, consider drawing boxes to help organize your work.[br][br][math]9\left(4x-3y-\frac{2}{3}\right)[/math]
Use the distributive property to write an expression that is equivalent to the expression. If you get stuck, consider drawing boxes to help organize your work.[br][br][math]-2\left(-6x+3y-1\right)[/math]
Use the distributive property to write an expression that is equivalent to the expression. If you get stuck, consider drawing boxes to help organize your work.[br][br][math]\frac{1}{5}\left(20y-4x-13\right)[/math]
Use the distributive property to write an expression that is equivalent to the expression. If you get stuck, consider drawing boxes to help organize your work.[br][br][math]8\left(-x-\frac{1}{2}\right)[/math]
Use the distributive property to write an expression that is equivalent to the expression. If you get stuck, consider drawing boxes to help organize your work.[br][br][math]-8\left(-x-\frac{3}{4}y+\frac{7}{2}\right)[/math]
Kiran wrote the expression [math]x-10[/math] for this number puzzle: “Pick a number, add -2, and multiply by 5.”[br][br]Lin thinks Kiran made a mistake. How can she convince Kiran he made a mistake?
What would be a correct expression for this number puzzle?
The output from a coal power plant is shown in the table:
Similarly, the output from a solar power plant is shown in the table:
Based on the tables, is the energy output in proportion to the number of days for either plant? If so, write an equation showing the relationship. If not, explain your reasoning.[br]