### Mathematics in Context: Patterns and Figures

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It goes on to introduce certain concrete and practical as well as abstract and theoretical aspects of mathematics, which have an essential place in the teaching and learning of the subject. The mathematics curriculum is organized under subject domains. The curriculum defines five basic areas of skill across all subjects and at all grade levels. For mathematics, these skill areas include the following:. Sub Menu.

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This involves identifying the key pieces of information needed to find the answer. This may require students to read the problem several times or put the problem into their own words. Sometimes you can solve a problem just by recognizing a pattern, but more often you must extend the pattern to find the solution.

Making a number table can help you see patterns more clearly. To use this strategy successfully, you need to be sure the pattern will really continue. Have students give reasons why they think the pattern is predictable and not based on probability.

## Illustrative Mathematics

Problems that are solved most easily by finding a pattern include those that ask students to extend a sequence of numbers or to make a prediction based on data. In this problem, students may also choose to make a table or draw a picture to organize and represent their thinking. Start with the top layer, or one basketball. Determine how many balls must be under that ball to make the next layer of a pyramid. Use manipulatives if needed. Students can use manipulatives of any kind, from coins to cubes to golf balls. Students can also draw pictures to help them solve the problem.

You may want to have groups use different manipulatives and then compare their solutions to determine whether the type of manipulative affected the solution. If students are younger, start with three layers and discuss their answers to this simpler problem.

## Engaging Students in Mathematics: Why Context is Critical

Then move on to more layers as students gain understanding of how to solve the problem. If it helps to visualize the pyramid, use manipulatives to create the third layer.

Record the number and look for a pattern. The second layer adds 3 basketballs and the next adds 5 basketballs. Each time you add a new layer, the number of basketballs needed to create that layer increases by 2. Then add the basketballs used to make all six layers. The answer is 91 balls. Look at the list to see if there is another pattern.

The number of balls used in each level is the square of the layer number. Determine if the best strategy was chosen for this problem, or if there was another way to solve the problem. Students should explain their answer and the process they went through to find it. It is important for students to talk or write about their thinking. Demonstrate how to write a paragraph describing the steps students took and how they made decisions throughout the process.

First, I started with the first layer. I used blocks to make the pyramid and made a list of the number of blocks that I used.

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Then I created a table to record the number of balls in each layer. I made four layers and then saw a pattern. I saw that for each layer, the number of balls used was the number of the layer multiplied by itself. I finished the pattern without the blocks, by multiplying the number of balls that would be in layers 5 and 6.

A woman is trying to cut down the number of cans of soda she drinks each week. She makes a plan so that in several weeks she will be drinking only one can of soda.

### Problem Solving: Find a Pattern

If she starts with 25 cans the first week, 21 cans the second week, 17 cans the third week, 13 cans the fourth week, and continues this pattern, how many weeks will it take her to reach her goal? Have students work in pairs, in groups, or individually to solve this problem. They should be able to tell or write about how they found the answer and justify their reasoning. Calculate with positive and negative integers, decimal numbers, fractions, and percentages; find common denominators; use a spreadsheet for simple calculations and presentations; justify solution methods; explore and describe structures and changes in numerical and geometrical patterns using figures, words, and formulas; formulate and solve simple equations; and simplify expressions with parentheses, addition, subtraction, and multiplication of numbers.

Analyze characteristics of two- and three-dimensional figures, build three-dimensional models, and draw simple three-dimensional figures in perspective; describe and perform reflection, rotation, and translation; describe position and movement in a coordinate system; and calculate distances parallel to the coordinate axes. Select suitable measurement tools, choose suitable units, and convert between units; explain the structure of measurements of length, area, and volume and calculate circumference, area, surface area, and volume of simple two- and three-dimensional figures; use a scale to calculate distances from a map and to make a scale drawing; use ratios in practical situations; calculate velocity; and convert currencies.

Collect data from observations, questionnaires, and experiments; represent data in tables and graphs digitally and manually; read, interpret, and assess data; find median, mode, and average for simple data sets, and assess them in relation to each other; assess and discuss probability in everyday contexts, games, and experiments; calculate probability in simple situations.

Investigate and describe characteristics of two- and three-dimensional figures, and use them for constructions and calculations; carry out, describe, and explain constructions with a compass and a ruler, and with a dynamic geometry program; use and explain the use of congruence and the Pythagorean theorem to calculate unknown lengths and angles; make and interpret scale drawings and perspective drawings; use coordinates to represent and investigate geometric figures; formulate logical reasoning about geometrical ideas.