Grade 7 / Literal expressions: introduction 51 exercises corrected

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ChingQuizz : 7 exercises available for Quizz assessment : 1. Problem situations E.5764 We would like to make square frames of photographs by decorating the circumference with small tiles of mozaic. Here are three examples of such frames : How many tiles are needed to form a frame with one side made of 6 tiles? 7 tiles? 14 tiles? E.5765 Using small mozaic tiles, we wish to pave squares as follows : horizontal edges are formed by a two-tile band ; the vertical edges are formed from a strip of one tile. Here is the making of two of his frames : Determine the number of tiles needed to pave a square whose side is made up of 8 small tiles? of 9 small tiles? of 14 small squares? E.5766 We wish to represent the ˇ skeleton ı of a cube by completing its edges with small cubes. Below are given two realizations of such ˇ skeletons ı : Determine the required number of ˇ small cubes ı to make the ˇ squelette ı of a cube whose edge measures 10 small cubes? 11 small cubes? 20 small cubes? 2. Introduction to literal expressions E.1812 In this exercise, we consider four quadrilaterals all having two sides of the same measure ; the measure of the other two varies between each quadrilateral: 1 Complete the table below : Quadrilatère ABCD EFGH IJKL MNOP x : : : : : : cm : : : : : : cm : : : : : : cm : : : : : : cm Périmètre : : : : : : cm : : : : : : cm : : : : : : cm : : : : : : cm Aire : : : : : : cm 2 : : : : : : cm 2 : : : : : : cm 2 : : : : : : cm 2 2 a From the previous table, place in the reference frame below four points representing the perimeter of each quadrilateral where : The abscissa is the value of x , The ordinate is the value of the perimeter (with the precision allowed by the marker) https://chingmath.fr quizzChap/1013 chapExoCorrec/5764 sacados/5764 Côté forméde3carreauxCôté forméde4carreauxCôté forméde5carreaux chapExoCorrec/5765 sacados/5765 Côté forméde5carreauxCôté forméde6carreauxCôté forméde7carreaux chapExoCorrec/5766 sacados/5766 Cube dont l’arêtemesure8petits cubesCube dont l’arêtemesure9petits cubes chapExoCorrec/1812 sacados/1812 ABCDxxcm2cmEFGHxx;5cmIJKLxxcmMNOPxx;5cm
b What do you notice? c Do the same for the points having for abscissa the val-ues of x and for ordinate the corresponding area. 3 Consider the expression 2 × x +4 . For each value assigned to x , give the value of the expression 2 × x +4 when you change x by its value. Complete the table below : Value of x 1 1.5 2 2.5 3 Value of 2 × x +4 E.1230 1 Consider the following three segments : We vary the length of the segments of length x . Deter- mine some values of the segments [ AB ] , [ CD ] , and [ EF ] as a function of the value of x by completing the table below : Value of x 0 1 2 10 AB CD EF 2 Connect each phrase below to the segment with the cor-responding length : a 12 − x b 3 × x + 12 c 2 × (12 + x ) E.4647 We consider squares made of small squares all identical. The small squares surrounding the fig-ure are shaded. Here are the first five figures constructed in this way: 1 For each of the five figures above, give the number of shaded squares making up the figure. 2 Give the number of grayed-out squares composing this type of figure when the side of such a figure is composed of 10 small squares. 3 Noting n the number of small squares composing the side of such a figure, give a formula for obtaining the number of small shaded squares present in this figure. E.1813 Here are the 32 calculations a teacher left for you to do the next day 2 × 4 + 1 ; 2 × 5 + 1 ; . . . ; 2 × 34 + 1 ; 2 × 35 + 1 Find the simplest statement summarizing this exercise. (We’ll imagine the instruction to be given by phone to a class-mate so that he or she can do all these calculations.) 3. Calculation Using an Expression E.1816 1 Find the value of the following expression for x =1 : A = x × 2 + 3 + x × x 2 Evaluate the following expression for x =2 : B = (2 + x × 3) × x E.564 Evaluate the following expressions for x =2 : a 3 × x + 2 b 2 × (3 − x ) − 1 c ( x + 2) × (5 − x ) E.1879 Evaluate each of the following ex-pressions for x =3 : a 3 × x + ( x − 2) × (2 × x + 1) b (2 × x − 1) × 2 + 3 E.10125 1 Evaluate each of the expressions : a 3 × (2 × x + 1) b 6 × x + 3 for the following three values : x =0 ; x =2 ; x =10 2 Can you justify the equality of these two literal expres-sions in each of these three cases? E.10924 For a =4 and b =5 , the following expressions have the values : b − 4 × a = : : : ; b × a − 4 = : : : https://chingmath.fr 01232468AbscisseOrdonnée chapExoCorrec/1230 sacados/1230 ABx12cm DCx12cm EF12cmx chapExoCorrec/4647 sacados/4647 Fig.1Fig.2Fig.3Fig.4Fig.5 chapExoCorrec/1813 sacados/1813 chapExoCorrec/1816 sacados/1816 chapExoCorrec/564 sacados/564 chapExoCorrec/1879 sacados/1879 chapExoCorrec/10125 sacados/10125 chapExoCorrec/10924 sacados/10924
E.10925 For c =2 , d =10 , e =14 , the follow-ing expression has the value : e + c d − c = : : : E.11703 Calculer les expressions suivantes pour x =3 : a 3 × x − 5 b x − 1 × 2 × x + 4 E.11705 Calculer les expressions suivantes pour x =3 . On donnera le résultat sous la forme d’une fraction simplifiée : a 5 × x + 3 2 × x + 2 b 3 × x − 4 4 × x − 6 + x + 2 6 4. Generating Literal Expressions: Geometry E.1233 1 For x worth 2 cm , give the length of segments [ AB ] , [ CD ] and [ EF ] . 2 Express the length of segments [ AB ] , [ CD ] and [ EF ] in terms of ˇ x ı. E.1229 The resulting literal expressions should be simplified to the maximum. Express, in each case, the length of the segments [ AB ] , [ CD ] , and [ EF ] in terms of ˇ x ı : a b c E.10126 The resulting literal expressions must fi be simplified to the maximum. 1 Express the area of the rectangle GHIJ and the perime-ter of KLMN in terms of x . 2 For x =3 cm , calculate the area of the rectangle GHIJ and the perimeter of KLMN E.1228 Express the perimeter and area of the rectangles ABCD and EFGH in terms of x . E.1227 The figure shown here is made up of rectangles. For each of the literal expressions below, specify whether these formulas represent a length, a perimeter, or an area: 1 For each question, draw a rectangle whose area is ex-pressed in terms of x : a 3 × 2 b 3 × x c ( x + 2) × 3 2 For each question, draw a rectangle whose perimeter is expressed in terms of x : a 2 × x + 2 × 3 b 2 × ( x + 5) c 2 × ( x + 3) E.11701 1 For x worth 2 cm , give the length of segments [ AB ] , [ CD ] and [ EF ] . 2 Express the length of segments [ AB ] , [ CD ] and [ EF ] in terms of ˇ x ı. https://chingmath.fr chapExoCorrec/10925 sacados/10925 chapExoCorrec/11703 sacados/11703 chapExoCorrec/11705 sacados/11705 chapExoCorrec/1233 sacados/1233 ABx7cm DCx12cm EFx2×x1cm1cm chapExoCorrec/1229 sacados/1229 ABx3cm CDx100m EF2kmx10km chapExoCorrec/10126 sacados/10126 GHIJ2cmxKLMNx2×x2cm chapExoCorrec/1228 sacados/1228 ABCD2×x3cmEFGHxx2cm chapExoCorrec/1227 sacados/1227 x2cmB3cmACDEF chapExoCorrec/11701 sacados/11701 ABx7cm DCx12cm EFx2×x1cm1cm
5. Literal Expression Evaluation: Computation Program E.11638 On considère le programme de calcul ci-dessous : 1 Lorsque la valeur d’entrée est 2 , quelle est la valeur de sortie de ce programme de calcul. 2 Lorsque la valeur d’entrée est 5 , quelle est la valeur de sortie de ce programme de calcul. E.11671 The following calculation program is considered : 1 Noting x the number chosen as the input to this compu-tational program, give the algebraic expression obtained as the output of this computational program. 2 By solving an equation, determine the number chosen as input so that this computation program returns the value 13 . E.11672 The figure below gives a schematic of a calculation program. 1 If the starting number is 1 , show that the result is − 15 . 2 If we choose any number x as the starting number, which of the following expressions gives the result obtained by the calculation program? Justify. A = x 2 − 5 × 3 x +2 B =(2 x − 5)(3 x +2) C =2 x − 5 × 3 x +2 3 Lily claims that the expression : D =(3 x +2) 2 − ( x +7)(3 x +2) gives the same results as the expression B for all values of x . Is Lily’s statement true? Justify. E.11702 The following calculation program is considered : 1 Noting x the number chosen as the input to this compu-tational program, give the algebraic expression obtained as the output of this computational program. 2 By solving an equation, determine the number chosen as input so that this computation program returns the value 13 . 6. Literal Expression Generation: Pattern Recognition E.1231 For each of the two cases below find the literal expression that was entered into the calculator to obtain the following table : 1 Valeur de x Résultat affiché 4 16 5 20 6 24 7 28 2 Valeur de x Résultat affiché 3 7 4 9 5 11 6 13 E.1232 In each of the two tables, a literal expression as a function of ˇ x ı was used to pass in each row from the column ˇ value of x ı to the column ˇ value of the expression ı. 1 Valeur de x Valeur de l’expression 1 10 2 17 3 24 4 31 2 Valeur de x Valeur de l’expression 0 2 1 3 2 6 10 102 https://chingmath.fr chapExoCorrec/11638 sacados/11638 Choisir un nombreMultiplier 3Ajouter2Soustraire 5Multiplier les deux nombres chapExoCorrec/11671 sacados/11671 Choisir un nombreAjouter 2Multiplier par4Multiplier par3Ajouter les deux nombres chapExoCorrec/11672 sacados/11672 Choisir un nombreCalculer son tripleCalculer son doubleAjouter2Multiplier les deux nombres obtenusSoustraire5 chapExoCorrec/11702 sacados/11702 Choisir un nombreAjouter 2Multiplier par4Multiplier par3Ajouter les deux nombres chapExoCorrec/1231 sacados/1231 chapExoCorrec/1232 sacados/1232
7. Being a solution to an equation E.1238 Check if the numbers 1, 3 and 5 are solutions of the equation : 3 x − 3 = 2 x + 2 E.10130 Check if the numbers 1, 3 and 5 are solutions of the equation : 4 x − 3 = 3 x + 2 E.1862 Definition: An equation is an equality that contains one or more unknown numbers, represented by letters. We say that’a number is a solution d’to an equa-tion if, when we substitute that number for the letter, the’equality becomes true. Check whether the numbers 2 and 3 are solutions to the equa-tion : 3 × x − 2 = 2 × x E.1878 Check whether the numbers 1 and 3 are solutions of the equation : 2 x + 1 = 10 x − 7 E.10127 Check if the numbers 1, 3 and 5 are solutions of the equation : 2( x + 1) + 3 x = 5 x + 2 E.1831 Consider the equation with two un-knowns x and y : y = 3 × x + 2 1 Find the value of y for the following values of x : a x =0 b x =1 c x =1 ; 5 2 Find the value of x that satisfies the equation for the following values of y : a y =2 b y =8 c y =9 ; 5 E.10928 The literal expressions obtained must be simplified as much as possible. 1 Express the area of the rectangle GHIJ and the perime-ter of KLMN in terms of x . 2 For x =3 cm , calculate the area of rectangle GHIJ and the perimeter of KLMN . Do an exercise with 2 × x + 2 × 2 × x + 2 × 2 6 x + 4 E.10929 1 For x equal to 2 cm , give the lengths of segments [ AB ] , [ CD ] , and [ EF ] . 2 Express the lengths of segments [ AB ] , [ CD ] , and [ EF ] in terms of ˇ x ‘’. x = 1 2 × : : : + 7 x = 2 2 × : : : + 7 x = 3 : : : x = 4 ldots x = x ldots E.11707 L’équation 4 × x +2=18 admet pour so-lution : 2 3 4 5 Indiquer la bonne réponse et justifier E.11708 L’équation 3 × x +1=2 × x +6 admet pour solution : 4 2 5 3 Indiquer la bonne réponse et justifier 8. Around equations E.10919 Solve the following equations : a x + 5 = 12 b x − 3 = 2 c 3 × x = 4.2 d x ÷ 5 = 1.2 https://chingmath.fr chapExoCorrec/1238 sacados/1238 chapExoCorrec/10130 sacados/10130 chapExoCorrec/1862 sacados/1862 chapExoCorrec/1878 sacados/1878 chapExoCorrec/10127 sacados/10127 chapExoCorrec/1831 sacados/1831 chapExoCorrec/10928 sacados/10928 GHIJ2cmxKLMNx2×x2cm chapExoCorrec/10929 sacados/10929 ABx7cm DCx12cm EFx2×x1cm1cm chapExoCorrec/11707 sacados/11707 chapExoCorrec/11708 sacados/11708 chapExoCorrec/10919 sacados/10919
Hint: we can use a diagram like below to solve these equa-tions : E.10920 Note: for the equation (2 × x )+1=7 , finding the solutions is easy, as all you need to do, using the priorities of the op-erations, to see how the expression of the left-hand member is constructed in order to obtain the numerical value of the right-hand member. The diagram below shows this resolution : The solution to the equation (2 × x )+1=7 is the number 3 . Determine the solution to the equation (7 × x )+3=17 by com-pleting the diagram below : E.11011 1 Complete the lower part of the following diagram: 2 What value of ˇ x ı verifies the equality below : (2 × x ) − 3 = 5 E.11033 Without justification, give the so-lutions of the following equations : a (3 × x ) + 1 = 4 b (5 × x ) − 4 = 6 c (2 × x ) + 1 = 5 E.11074 Using for each question a diagram similar to the one above, find the value of ˇ x ı that verifies the equality: a (3 × x ) − 2 = 7 b (5 × x ) + 10 = 16 c (3 × x ) + 1 = 5.5 d 2 × (3 × x ) + 1 = 8 E.11704 En justifiant votre démarche, résoudre les équations suivantes : a 4 × x + 7 = 18 b 5 × x − 1 = 1 Indication : on pourra utiliser un diagramme ressemblant à: 9. Around the equations (two side) E.11075 Without justification and in each case, give the value of x allowing the balance to be balanced : a b E.10921 For each question, determine the value of x that balances the scale : a b c d E.11076 Without justification and in each case, give the value of x allowing the balance to be balanced : a b 10. The Square: Calculating with an Expression E.10922 1 Complete the following dotted lines : a 2 + 5 = : : : b 2 × 5 = : : : c 5 2 = : : : https://chingmath.fr ???? chapExoCorrec/10920 sacados/10920 x2×x(2×x1763×2÷2−1 x17 chapExoCorrec/11011 sacados/11011 x2×x(2×x−35×2−3÷2 chapExoCorrec/11033 sacados/11033 chapExoCorrec/11074 sacados/11074 chapExoCorrec/11704 sacados/11704 x chapExoCorrec/11075 sacados/11075 x41xx x43x6x chapExoCorrec/10921 sacados/10921 4×x33×x5 5×x12×x10 8×x24×x14 3×x2x6 chapExoCorrec/11076 sacados/11076 x212x 72x3x3 chapExoCorrec/10922 sacados/10922
2 Complete the following dotted lines : a 2 + 6 = : : : b 2 × 6 = : : : c 6 2 = : : : E.10923 Complete the following dotted lines : a 10 2 = : : : b 2 2 = : : : c 7 2 = : : : d 0 2 = : : : e 3 2 = : : : f 5 2 = : : : E.10926 For a =4 , b =7 , c =3 , the following expressions have the values : a 2 − b 2 − 4 − c 2 = : : : ; b 2 − a 2 − c 2 = : : : 11. The square: being the solution to an equation E.10128 Check whether the numbers 3 and 7 are solutions of the equation : x 2 − 3 x + 16 = 7 x − 5 E.10129 Check if the numbers 1, 3 and 5 are solutions of the equation : x 2 + 1 = 2 x E.10131 Check whether the numbers 1 and 4 are solutions of the equation : x 2 + 3 = 5 x − 1 12. Unclassified exercises E.11706 Voici les premières étapes d’un motif répétitif: 1 Combien de carreaux est composée la figure à l’étape 4? 2 Pour n un entier supérieur ou égal à 1 , exprimer le nom-bre de carreaux nécessaires en fonction de n . https://chingmath.fr chapExoCorrec/10923 sacados/10923 chapExoCorrec/10926 sacados/10926 chapExoCorrec/10128 sacados/10128 chapExoCorrec/10129 sacados/10129 chapExoCorrec/10131 sacados/10131 chapExoCorrec/11706 sacados/11706 Étape 1Étape 2Étape 3