دليل المعلم Volume 2 الرياضيات للصف العاشر منهج انجليزي Reveal الفصل الثالث 2021 2022
Standard
1 Make sense of problems and in solving
them Mathematically proficient students start explaining to themselves the meaning Of a problem and loking f0r entry points to its They analyze givens, cmstraints. relationships, and goals. They make cmjectures atmJt the and meaningof the solubon and plan a solution pathway rather than sinwy jumping into a sdution attempt. They consider problems, and try special cases and simpler forms ot the original problem order to gain into its solution. They monitor and evaluate their prowess md change course if neceswy. Older students might dependng on the context of the problem. transform expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students cm explain corresw•.dencesl between equations, verbal descriptions. tables, and graphs Or draw diagrams Of important features and relationships graph data. and search for regularity trends. students mi't rely on using concrete otects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different methcxl. they cmtinually ask themselves. -Does this mde sense?" They can understand the ap of others to solving complex problems and identify correspondences between different
2 Reason quantitatively
Mathematically proficient students make sense Of quantities and twir relationships in moblem situations They bring two complementary abdities to tmr on gyoblems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbds as if they have a life of their cywn, without necessarily attendng to their referents—and the abdity to contextudize. to pause as needed during the manipulation in to into the referents for the symbds molved. Quantitative reasoning entails habts of creating a coherent representation of the problem at hand: considering the units involved: attendng to the meaning of cwantities, not just how to compute them: and krmng flexby using different of and
3 Construct viable arguments and critique the reasoning of otters
Mathematically proficient students understand and use stated assumptions, defhitions. md creviously established results in constructing arguments. They make conjectures and build a logical Of statements to exgore the truth of their conjectures. They are able to analyze situations by breaking them into cases. and cm and use cmmterexamples. They justify their conclusions, communicate them to Others. and respond to the arguments of others. They reason inductively about data. making plausible arguments that take into the context from which the data arose. Mathematically proficient students Me also able to tie effectiveness two plausible arguments. dstinguish correct reasoning from that which is flawed. Mdif there is a flaw in an argument—expl ain what it is. Elementary students can cmstruct arguments using concrete referents such as Objects, drawings, diagrurs, and acticxts. Such arguments can make sense and correct. even though they Me not generaliz«i formal unti later Later. students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments Of others, dedde whether they make sense, md ask useful westims to clarify or improve me arguments
4 Model with mathematics
Mathematically proficient students can a»ly the mathematics they know to solve problems arising in everyday life. and the workplace. In early grades. this might be as simple as writing addition equation to
a situation. In middle grades. a student might apply prcortional reasoning to plan a event or analyze a problem in community. By high sctml. a student might use georr.etry to save a design problem use a
function to descrit how quantity of interest ckpends on another. Mathematically proficient students who cm apply what they know are comfortable making assumptions and approximations to simplify a complicated situatim
redizing that these may need revision later. They are able to identify important quantities in a practical situatim and map their relationships using such as two -way tables, graphs, flowcharts and formulas. They Can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical resußts in the context of the and reflect on whether Ole results make sense. improving the if t has served its
Standards for Mathematical Practect Reveal Math
Standard
Use appropriate tools strategically Mathematically proficient students consider the available tools when sdving a mathematical problem. These tods might include pencil and concrete models, a ruler. a protractor, a cdculator. a spreadsheet. a computer algebra systern, a statistical package, tynamic geometry software. Proficient students are sufficiently tamiiar with tools appropriate for their gra or course to make sound decisions about when each of these tools might be helpful, recowlizing both the insight to be gained and their limitations For example, mathematically proficient high school students analyze graphs Of functions and solutions generated using a graphing calculator. They detect possble errors by strategically using estimation and other mathematical knowledge. When making mathematical models. they know that technology enaNe them to visualize the results of varying assumptions. explore
consequences. and compare predictions with data. Mathematically proficient students at various grade levels Me able to relevant externa mathematical resources, such as digital content located on a website, use them to pose or solve problems. They are able to use technological tools to expm and deepen their understandng of cmcepts
Attend to precision
Mathematically proficient students try to cM1municate precisely to others. They try to use clear definitions in dscussion others and in their cmn reasoning. They state the meaning of the symbols they choose, induding usülg the equal sign consistently and appropriately. They are careful about specifying units of measure. and labeling axes to clarity the correspondence with quantities in a They calculate accurately and efficiently, I exVess numerical answers with a degree of precisicn appropriate for the problem context In the elementary yam students give carefully formulated explanations to eachother. By the the they reach have learned to examne dims and rnüe explicit use Of definitms
7 Look for and make use Of structure
Mathematically proficient students look closely to discern a pattern or structure. Young students. for example, might notice that three md seven mue is the same unount as seven and three more. Or they may sort a cdlectim of shapes acccyding to how many sides the shapes have. Later. students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expressim xa + 9x + 14, older students can see the g as 2 x 7 and the 9 as 2 + 7. They recognize the significance of existing in a geometric figure and can use the strategy of drawing an auxiliary line for sowing problems. They also can step back for an werview and shift perspective. They can see complicated things, such as some algebraic expressions, as single octs as bemg composed of several objects. For examøe. they see 5 — — y'2 as 5 milus a positive number times a square and use that to realize that its value cannot be more than 5 fcy any real numbers x and y
Look for and express regtAarity in repeated reasoning
Mathematically proficient students notice if calculations are repeated. and look both for general methods and for shortcuts. Upper elementuy students notice when dividing 25 by 11 that they are repeating the cdculatims over and over again. and cmclude they have a repeating decimal. By paying attention to the cdculation of slope as they repeatedly check whether pc*nts are on the line thrwgh (1.2) with slope 3, middle school students might abstract the equatim (y— 2V(x — 1) = 3. Noticing the regularity in the way terms cancel when expanding (x— l)'x + 1), (x —' + x + 1). (x — + i + x + 1) might lead them to the genera formula for the sum of a geometric series. As they work to solve a problem. mathematically proficient students mintin overs.t of the process, while attending to the detads. They continually evaluate the reasonableness of their "'termediate results