Mathematics CMP3, Grade 7 Stretching and Shrinking Month: Nov-Dec, Weeks 11-14

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1 Standards for Mathematical Practice Critical Areas: Ratios and Proportions, Number System, Expressions and Equations Content Emphases for Grade 7 Major Cluster 70% of time Supporting Cluster 20% of Time Additional Cluster 10% of Time 7.RP.A.1,2,3 7.SP.A.1,2 7.G.A.1,2,3 7.NS.A.1,2,3 7.SP.C.5,6,7,8 7.G.B.4,5,6 7.EE.A.1,2 7.SP.B.3,4 7.EE.B.3,4 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning Domains: Clusters Ratios and Proportions Expressions and Equations Geometry 7.RP.A Analyze proportional relationships and use them to solve real-world and mathematical problems. 7.RP.A.2 Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in 7.EE.B Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.G.A Draw, construct, and describe geometrical figures and describe the relationships between them. Standards The standards in bold should drive the learning for this period of instruction. 7.EE.B.3 Solve multistep real-life and 7.G.A.1 Solve problems involving mathematical problems posed with scale drawings of geometric figures, positive and negative rational including computing actual lengths numbers in any form (whole and areas from a scale drawing and numbers, fractions, and decimals), reproducing a scale drawing at a using tools strategically. Apply different scale. properties of operations to calculate 7.G.A.2 Draw (freehand, with ruler with numbers in any form; convert and protractor, and with between forms as appropriate; and technology) geometric shapes with assess the reasonableness of given conditions. Focus on answers using mental computation constructing triangles from three and estimation strategies. measures of angles or sides, noticing when the conditions 7.G.B Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.B.6 Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Rev 7.16 PROPERTY OF MPS Page 1 of 10

2 tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. 7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. 7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. determine a unique triangle, more than one triangle, or no triangle. Prior work 6.NS.C.5,6 6.G.1,2,3 7.G.A.2 6.EE.A.2 6.EE.C Pathways to Algebra Readiness Future Work 7.RP.A.2 8.EE.4 8.F.A.2-3 Rev 7.16 PROPERTY OF MPS Page 2 of 10

3 Key Student Understandings 7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems. Students understand proportions are equivalent ratios. Students understand how to use proportional relationships to solve multistep problems. Use scale factors and ratios to describe relationships among the side lengths, perimeters, and areas of similar figures. Recognize the relationship between scale factor and ratio in similar figures Use scale factors or ratios to find missing side lengths in a pair of similar figures 7.G.A Draw, construct, and describe geometrical figures and describe the relationships between them. Students understand what it means for figures to be similar. Generalize properties of similar figures. Use informal methods, scale factors, and geometric tools to construct similar figures (scale drawings) 7.G.B Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. Identify similar figures by comparing corresponding sides and angles Explore the effect on the image of a figure if a number is added to the x- or y-coordinates of the figure s vertices Predict the ways that stretching or shrinking a figure will affect side lengths, angle measures, perimeters, and areas 7.EE Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Students understand how to construct simple equations to solve real-world problems Students understand the format y=kx represents a proportional relationship where k is the constant of proportionality. Distinguish algebraic rules that produce similar figures from those that produce nonsimilar figures. Use algebraic rules to produce similar figures Recognize when a rule shrinks or enlarges a figure Assessments Formative Assessment Strategies Evidence for Standards-based Grading Rev 7.16 PROPERTY OF MPS Page 3 of 10

4 Common Misconceptions/Challenges 7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems. Students often believe that all relationships involving multiplication are proportional. Proportional relationships must have a constant of proportionality. It may be somewhat of a challenge for both students and teachers alike to analyze proportional relationships and use them to solve problems without using cross-multiplication! 7.EE Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Students may struggle to identify which variable is the x (independent) and y (dependent). 7.G Draw, construct, and describe geometrical figures and describe the relationships between them. Students may struggle with scale when the scale factor is not a whole number. Students may want to divide for reductions rather than multiply by a fraction. Students may need explicit instructions that a scale factor of 1 produces the same image as the original, that less than one is a reduction, and more than 1 in an enlargement. Instructional Practices Note: This is not the time for students to learn to cross multiply to solve proportions. In 6th grade, RP standards talk about using ratio reasoning to solve problems. In 7th grade, the RP standards speak of analyzing proportional relationships and using them to solve problems. Cross multiplication is a procedure which shortcuts reasoning and analysis so it runs counter to the philosophy expressed in the RP standards. Moreover, when one reads the individual RP standards, there is no mention of anything that would require cross-multiplication. The CCSSM ratio and proportion standards stress understanding, reasoning, and analysis of ratios and proportional relationships. They do not mention, and do not require that students learn, the crossmultiplication procedure. Domain: 7.RP Ratios and Proportional Reasoning 7.RP. A Analyze proportional relationships and use them to solve real-world and mathematical problems. 7.RP.A.2 Examples: A student is making trail mix. Create a graph to determine if the quantities of nuts and fruit are proportional for each serving size listed in the table. If the quantities are proportional, what is the constant of proportionality or unit rate that defines the relationship? Have student explain how they determined the constant of proportionality and how it relates to both the table and graph during the summary of the lesson. Rev 7.16 PROPERTY OF MPS Page 4 of 10

Serving Size 1 2 3 4 Cups of Nuts (x) 1 2 3 4 Cups of Fruit (y) 2 4 6 8 The relationship is proportional (7.RP.2a).

5 Serving Size Cups of Nuts (x) Cups of Fruit (y) The relationship is proportional (7.RP.2a). For each of the other serving sizes there are 2 cups of fruit for every 1 cup of nuts (2:1). The constant of proportionality (7.RP.2b) is shown in the first column of the table and by the point (1,r) on the graph (7.RP.2d). The table below gives the price for different numbers of books. Do the numbers in the table represent a proportional relationship? Explain. Number of Books Price Solution: (7.RP.2a) Students can examine the numbers to determine that the price is the number of books multiplied by 3, except for 7 books(7.rp.2b). The row with seven books for $18 is not proportional to the other amounts in the table; therefore, the table does not represent a proportional relationship. Students could graph the relationships to determine if two quantities are in a proportional relationship and to interpret the ordered pairs. If the amounts from the table above are graphed (number of books, price), the pairs (1, 3), (3, 9), and (4, 12) will form a straight line through the origin (0 books, 0 dollars), indicating that these pairs are in a proportional relationship. The ordered pair (4, 12) means that 4 books cost $12. However, the ordered pair (7, 18) would not be on the line, indicating that it is not proportional to the other pairs. The ordered pair (1, 3) indicates that 1 book is $3, which is the unit rate, but does not apply to (7,18). (7.RP.2d) The graph below represents the cost of gum as $2 per pack of gum. Represent the relationship using a table and an equation. (7.RP.2c) Table: Number of Packs of Gum (g) Cost in Dollars (d) Rev 7.16 PROPERTY OF MPS Page 5 of 10

6 Equation: d = 2g, where d is the cost in dollars and g is the packs of gum 7.RP.A.3 Examples: You have 36 pictures to print for your scrapbook. In 5 minutes, your printer prints 10 pictures. If the printer continues to print at the same rate, how many minutes will it take to print 36 pictures? Use a table and a graph to illustrate your solution. One lap around a dirt track is 1 mile. It takes Bryce 1 hour to ride one lap. What is Bryce s unit rate, in miles, around the track? 3 9 Domain: Expressions and Equations Cluster 7.EE.B Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.B.4a It cost $1155 to send 33 packages through a certain shipping company. Consider the number of packages per dollar. Write an equation to represent the relationship. During summer vacation, Lydie spent time with her grandmother picking blackberries. They decided to make blackberry jam for their family. Her grandmother said that you must cook the berries until they become juice and then combine the juice with the other ingredients to make the jam. a. Use the table below to determine the constant of proportionality of cups of juice to cups of blackberries. b. Write an equation that will model the relationship between the number of cups of blackberries and the number of cups of juice. Solutions: k=1/3 and therefore j = 1/3 b Domain: Geometry Cluster 7.G.A Draw, construct, and describe geometrical figures and describe the relationships between them. 7.G.1 Rev 7.16 PROPERTY OF MPS Page 6 of 10

7 A rectangular pool in your friend s yard is 150 ft. 400 ft. Create a scale drawing with a scale factor of 1 you computed the scale drawing lengths Use a table or an equation to show how 7.G.A.2 Students draw geometric shapes with given parameters. Parameters could include parallel lines, angles, perpendicular lines, line segments, etc. Example 1: Draw a quadrilateral with one set of parallel sides and no right angles. Students understand the characteristics of angles and side lengths that create a unique triangle, more than one triangle or no triangle. Rev 7.16 PROPERTY OF MPS Page 7 of 10

8 Example 2: Can a triangle have more than one obtuse angle? Explain your reasoning. Example 3: Will three sides of any length create a triangle? Explain how you know which will work. Possibilities to examine are: a. 13 cm, 5 cm, and 6 cm b. 3 cm, 3cm, and 3 cm c. 2 cm, 7 cm, 6 cm Solution: A above will not work; B and C will work. Students recognize that the sum of the two smaller sides must be larger than the third side. Cluster: 7.G.B.Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. 7.G.B.6 Students continue work from 5th and 6th grade to work with area, volume and surface area of two-dimensional and three-dimensional objects. (composite shapes) Students will not work with cylinders, as circles are not polygons. At this level, students determine the dimensions of the figures given the area or volume. Know the formula does not mean memorization of the formula. To know means to have an understanding of why the formula works and how the formula relates to the measure (area and volume) and the figure. This understanding should be for all students. Surface area formulas are not the expectation with this standard. Building on work with nets in the 6th grade, students should recognize that finding the area of each face of a three-dimensional figure and adding the areas will give the surface area. No nets will be given at this level; however, students could create nets to aid in surface area calculations. Students understanding of volume can be supported by focusing on the area of base times the height to calculate volume. Students solve for missing dimensions, given the area or volume. Students determine the surface area and volume of pyramids. Volume of Pyramids Students recognize the volume relationship between pyramids and prisms with the same base area and height. Since it takes 3 pyramids to fill 1 prism, the volume of a pyramid is 1/3 the volume of a prism (see figure below). Rev 7.16 PROPERTY OF MPS Page 8 of 10

9 To find the volume of a pyramid, find the area of the base, multiply by the height and then divide by three. V = Bh B = Area of the Base 3 h = height of the pyramid Example 1: A triangle has an area of 6 square feet. The height is four feet. What is the length of the base? Solution: One possible solution is to use the formula for the area of a triangle and substitute in the known values, then solve for the missing dimension. The length of the base would be 3 feet. Example 2: The surface area of a cube is 96 in2. What is the volume of the cube? Solution: The area of each face of the cube is equal. Dividing 96 by 6 gives an area of 16 in2 for each face. Because each face is a square, the length of the edge would be 4 in. The volume could then be found by multiplying 4 x 4 x 4 or 64 in 3. Differentiation 7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems. Use of multiple representations, such as double number lines, tape diagrams, tables, and graphs. 7.EE Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Use tables, graphs, and unit rates to help students write equations in the form of y = kx. 7.G Draw, construct, and describe geometrical figures and describe the relationships between them. Use concrete representations such as tools, manipulatives, drawings that demonstrate scale. Challenges Ask students to explain their thinking and ask questions such as What would happen if? Literacy Connections Academic vocabulary terms Vocabulary Strategies Literacy Connections ELL Strategies for Support Rev 7.16 PROPERTY OF MPS Page 9 of 10

10 Students are offered projects to extend their understanding. Students create story problems to model concepts in unit. Assign Extension problems in the A.C.E. section of CMP3 for current or related investigations. CMP3 Stretching and Shrinking Instructional Resources Developing Fluencies Conceptual Understanding and Fluency Games Rev 7.16 PROPERTY OF MPS Page 10 of 10

Compute fluently with multi-digit numbers and find common factors and multiples.

Academic Year: 2014-2015 Site: Stork Elementary Course Plan: 6th Grade Math Unit(s): 1-10 Unit 1 Content Cluster: Compute fluently with multi-digit numbers and find common factors and multiples. Domain: