Scale is a concept that is found at numerous grade levels in the Nova Scotia Mathematics curriculum. Scale drawings and models, similar polygons, and proportions are all found in mathematics outcomes. In math, scale is the ratio of the length in an image (or model) to the length of the actual object. Below is a question relating to scale factors. A scale factor is the ratio of any two corresponding lengths in two similar geometric figures. Take a look at the three different versions of Connect Four. Estimate the scale factor between each pair of game boards from the given pictures. Estimate the radius of each of the coloured chips. Is the scale factor of the radius of each coloured chip the same as the scale factor of their volume? You might ask students how scale is different from proportion. Try out this question: How big would a game board of Connect Four Hundred be (or even Connect Four Million) compared to Connect Four? In visual arts, scale refers to the size ratio between objects within an image. Using a consistent scale will make a drawing look more realistic. Objects do not appear too large or too small when compared to each other. Sometimes however, an artist might intentionally change the scale of certain objects in an image. One such technique is called 'Hieratic scale' or sometimes 'Hierarchical proportion'. This technique can be seen in paintings and sculpture from the middle ages where powerful or holy people were sometimes painted larger than ordinary or less important people to show their relative importance. The larger a person was, the greater their importance.
It would be fun to show students examples of how visual artists play with scale in order to make an impact on the viewer. Students might even be given an opportunity to create a piece of art that has an exaggerated or inconsistent scale or plays with forced perspective. Nova Scotia Mathematics Curriculum Outcomes Grade 6 N05 - Students will be expected to demonstrate an understanding of ratio, concretely, pictorially, and symbolically. Grade 8 N04 - Students will be expected to demonstrate an understanding of ratio and rate. Grade 8 N05 - Students will be expected to solve problems that involve rates, ratios, and proportional reasoning. Grade 9 G03 - Students will be expected to draw and interpret scale diagrams of 2-D shapes. Math at Work 10 G03 - Students will be expected to demonstrate an understanding of similarity of convex polygons, including regular and irregular polygons. Mathematics Essentials 11 D9 - calculate scale factors in 2-D scale diagrams and 3-D scale models understand the relationship among the scale factor and the related change in area or volume.Math at Work 11 G02 - Students will be expected to solve problems that involve scale. Mathematics 11 M03 - Demonstrate an understanding of the relationships among scale factors, areas, surface areas and volumes of similar 2-D shapes and 3-D objects. EL
High school exams in Nova Scotia (grade 10-12) typically run in the second to last week in June. This is the time that grade 9 students are writing some exams as well. The week prior to exams is often set aside by teachers for a cumulative review. This is a time to revisit the curriculum outcomes, consolidate learning and make final connections. What teachers do with this review time can vary greatly from class to class. Many teachers create a booklet of review questions that cover the main topics from the course. This can be a helpful resource for some students but not all that engaging. Here are some additional ideas for reviewing outcomes that might increase student engagement and allow for some differentiation.
No matter how you decide to review for the exam, by the time you get to exams, "the hay is in the barn". Here is a note from Alberta Education... "The best way to prepare students for writing the achievement test is to teach the curriculum well and to ensure that students know what is expected. Many of the skills and attitudes that support test writing are, in fact, good skills and strategies for approaching all kinds of learning tasks." EL
Coins in a Row is a simple game that I have used in lots of classrooms at different grade levels. It is a great way to tackle some mental math and logical reasoning. I first learned about this game in an article from Ivar Peterson's Math Trek. The game originally appeared in Peter Winkler's Mathematical Puzzles: A Connoisseur's Collection (2004). The Rules: Arrange a bunch of coins of various denomination in a row (any even number of coins will do). The first player chooses a coin from one of the ends and slide it over to her pile. Your opponent then chooses a coin from one of the ends of the row of remaining coins. You and your opponent take turns removing a coin in this manner until they are gone. The player with the highest total value of coins wins. The fun part of this game is that it is unfair. Unfair games are a great way to get out of doing some chores while you're visiting in-laws over the holiday break (e.g. "Lets play a game, the loser has to wash the dishes"). The player who goes first can always win if they know the correct strategy. After playing this game a few times, I let students know that it is an unfair game and ask them if they can find a winning strategy. Here is how the winning strategy works. Label the coins from 1 to n going from left to right. Add up the value of all the odd labeled coins then add up the value of all the even labeled coins. The first player can choose either an odd or even labeled coin. The second player only has two evens to choose from if the first player took the first coin from the row. The second player only has two odd coins to choose from if the first player to the coin from the end of the row. The first player then just takes a coin from whichever end the second player chose from and the first player can guarantee that they get either all the odd or even labeled coins. This strategy guarantees at least a tie for the first player. It is not necessarily the "best" strategy however. According to Ivars Peterson, no one has yet worked out an optimal strategy that works for any number of coins.
Alternate versions are quite easy to create for this game. For example, Coins in a Circle. In this version, the first player can pick any coin, then play continues as in the previous version. Another way to play would be to use a deck of cards (remove the face cards) and deal out 10 cards in a row. EL
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