|
Notes from Class:
Math Notes: Transformations Transformations are drawings. There are three types: translations, reflections and rotations. Translations: When a figure on a grid looks like it "slid" to another spot on the grid, it is said to have translated. The size and shape of the image is not altered. The orientation (how it sits on the page) is not changed either. The only thing that changes is the location. Because the image appears to "slide" to its new location, translations are sometimes referred to as "slides". Each point must move so many points to the right or left, and so many points up or down. For example: If we move an image 3 to the right and 4 down, we must make sure that each vertex, or corner, makes that move. Move and check each point individually. Reflections: A reflection is a mirror image of a shape. There is a "line of reflection" and that acts like the mirror. The points on the figure must go from one side of the line to the other, making sure each point is exactly the same distance from the line of reflection. Rotations: The key to this move is to know clockwise (to the right) and counterclockwise (to the left), and degrees and/or angles. Students are asked to rotate a shape clockwise or counterclockwise so many degrees (or a fraction of a turn). For example, if the shape needs to be rotated 90 degrees clockwise it spins 1/4 turn to the right. Probability Probability is the "chance" an event will occur. We take the number of possible outcomes and compare that with the outcome you wish to occur. If you roll a die, there are 6 possible outcomes (1,2,3,4,5 or 6). The chance of getting a 2 when we roll (the outcome we want) is 1 out of 6. We can write this as a fraction (1/6), a ratio (1:6) or a percent (17%). The chance of getting a 3 would be the same. The chance of getting a 2 or a 3 is 2 out of 6, because we can get two desired outcomes, but there are still only 6 possible. There are two types of probability: theoretical probability and experimental probability. Theoretical probability is what is "supposed to happen" while experimental probability is "what does happen". For example, if you flip a coin 10 times. the probability of getting heads is 1 out of 2 ( 5 times out of 10). This is theoretical probability. Of course, we all know that there will be times where that won't happen (experimental probability). You might get 7 heads with 10 flips of the coin. You should know the difference between the two types. Sample question Doug wanted to roll a die to see how often he would roll a 3. There are 6 sides to the die and only one 3. The theoretical probability that he will roll a 3 is 1 out of 6 (17%). He rolls the die 24 times and gets a 3 six times. The experimental probability that he would roll a 3 is 6 out of 24 (or 1 out of 4 or 25%). Often the experimental probability does not match the theoretical probability in one experiment. If Doug did the experiment numerous times he would likely find that the occurrence of the three would tend toward 1 out of 6. Data Management In this chapter we will review first hand and second hand data, bias, questioning, and the various ways to present information gathered from a data source. The grade will come from a project that students will complete. First Hand Data: This is information that you gathered yourself through actual experimentation. This is not data that you research and find from other sources. YOU ask the questions and collect the answers. Second Hand Data: This is information that others have found and you are borrowing it for your purpose. Stats Canada loans out information to businesses often. The businesses are using second hand data. Bias: prejudice or favoritism toward a particular idea, group, object, person, etc. Make sure your question does not lean people to respond a certain way. You want them to be honest and open. Also, make sure that your population that you are asking does not automatically effect the results. For example, do not go to an ice rink to find out what is the favorite sport. Project: Page 1 Title page: title, name, picture (optional) Page 2 Your original tally sheet: title, t-chart, tallies Page 3 Table of information: title, t-chart, totals (very neat) Page 4 Graph: title, graph, labels, neat, accurate, key, etc. (use a ruler). Volume Volume is the amount of space "inside" a 3-dimensional object. The objects we will be dealing with are pyramids and prisms. The concept involves trying to find out how many cubes (picture a game die) would fit inside the box. Unlike area, which has two dimensions (length and width), and is a flat shape, volume involves a third measurement, height or depth. There is a shortcut formula for volume of a prism (a rectangular box in our case) as well. It is the length and width of the bottom of the box multiplied together, and then times that answer by the height of the box. V= l x w x h Be careful to choose the correct numbers. Do not multiply two lengths or two widths, etc. The answer is stated in "cubic units" because we are theoretically counting the number of cubes that would fit in the box. Because there are three numbers to multiply, we place a little "3" next to the unit. 3 example: 34 cm Area Area is the amount of space covered by a 2 dimensional object. It is used when trying to find out how much carpet would be needed to cover a bedroom floor, or how many bundles of shingles would be needed to cover a roof. It is measured in "square" units (square meters, square centimeters, etc.). Our focus will be primarily on quadrilaterals that we recognize (rectangles, squares, parallelograms, etc.). The formula for all will be the same (Area equals length times width: A=lxw). The length would be the longer side and the width the shorter. Students should also consider Area equals base times height (A=bxh), the base being the bottom of the shape and height being how tall the figure is. The answer would be the same because the same two numbers will be multiplied in each case (just perhaps not in the same order). In grade seven (and beyond), the terms "base" and "height" will be introduced when dealing with triangles, volume of various shapes, etc., so it would do well for them to become familiar with these terms. Perimeter Perimeter is the distance around the outside of a polygon. We figure it out by adding up all the lengths of the sides. Make sure all the units (meters, centimeters, etc.) are the same. Convert any that are different before adding. Geometry Terms to remember: Line: has no beginning and no end. Line segment: has a beginning and an end. Ray: has a beginning but no end. Intersecting: when lines, line segments or rays connect (come together or cross). Parallel: Lines, line segments or rays that go side by side and never meet (and never will meet if extended). Perpendicular: Lines, line segments or rays that meet to form a 90 degree angle. Angle: The opening created when lines, line segments or rays intersect. Vertex (vertices): the corner where the angle is created. Degrees: the unit used to measure the size of an angle. Circle: a circle has 360 degrees. Acute: an angle measuring less than 90 degrees. Right: an angle measuring exactly 90 degrees. Obtuse: an angle measuring larger than 90 but less than 180 degrees. Straight: an angle measuring exactly 180 degrees. Reflex: an angle measuring more than 180 but less than 360 degrees. Adjacent: angles that share a ray, line or line segment (next to each other). Complimentary: angles that add up to 90 degrees. Supplementary: angles that add up to 180 degrees. More terms: Polygon: a closed in figure with three or more straight sides. Regular: a polygon with equal length sides. Triangles: a three sided polygon. Types of triangles: There are two ways to classify triangles. 1. The number of equal sides and angles. Scalene: a triangle with no equal sides or angle measures. Equilateral: a triangle with all equal sides and angle measures. Isosceles: a triangle with two equal sides and two equal angle measures. 2. The size of the largest angle. Acute-angled triangle: the largest angle is less than 90 degrees. Right-angled triangle: the largest angle is exactly 90 degrees. Obtuse-angled triangle: the largest angle is more than 90 degrees. Congruent: When angles or triangles are equal in every way. Quadrilaterals: a 4 sided polygon. Types of quadrilaterals: Parallelogram: a quadrilateral with two sets of equal sides and 2 sets of parallel sides. Rectangle: a quadrilateral with two sets of equal sides, 90 degree angles and 2 sets of parallel sides. Rhombus: a quadrilateral with 4 equal sides and 2 sets of parallel sides. Square: a quadrilateral with 4 equal sides, 2 sets of parallel sides and 90 degree angles. Trapezoid: a quadrilateral with one set of parallel sides. Kite: a quadrilateral with 2 sets of equal sides. Pentagon: a polygon with 5 sides. Hexagon: a polygon with 6 sides. Heptagon: a polygon with 7 sides. Octagon: a polygon with 8 sides. Nonagon: a polygon with 9 sides. Decagon: a polygon with 10 sides. Dodecagon: a polygon with 12 sides. Preservation of Equality This is a short "add-on" unit to the patterns and relations chapter. The first concept that students need to know is the concept of commutative property. When you add 2 + 4 you get the same answer as 4 + 2. Addition has "commutative property". You can switch the numbers around and the answer is still the same. Example: 6 + 8 = 8 + 6 Multiplication also has commutative property. Example: 9 x 3 = 3 x 9 Subtraction and division DO NOT have commutative property. Example: 12 divided by 3 is not the same as 3 divided by 12. 24 - 6 is not the same as 6 - 24. Preserving equality means to make sure the right and left side of the equals sign remains "equal". Use commutative property to preserve equality in this equation: 3 + 8 = ________ The answer would be 8 + 3. Use commutative property to preserve equality in this equation: 2 x 7 = ________ The answer would be 7 x 2. Preserving Equality by using Opposite Operation This strategy will eventually be used to help solve equations where an unknown variable is involved. The key is to remember to keep the equation balanced by preserving equality; what you do to one side of the equals sign you do to the other. Examples: n + 4 = 17 The opposite of "+ 4" is "- 4", so we use that operation. n + 4 - 4 = 17 - 4 What we do to one side, we do to the other to preserve equality. n = 13 The "+ 4" and "- 4" cancel each other (equal zero). Do 17 - 4. Because the value of "n" has not changed, the equation at the beginning (n + 4 = 17) is said to be equivalent to the equation at the end (n = 13). 3n - 5 = 1 Start with getting rid of the "- 5" by adding 5. 3n - 5 + 5 = 1 + 5 Keep the equation balanced to preserve equality. 3n = 6 The "- 5" and the "+ 5" cancel each other out. Add the 1 + 5 on the right. The value of the "n" does not change from the beginning (3n - 5 = 1) to the end (3n = 6), so the first equation and the last equation are said to be equivalent. Sample Question Use the preservation of equality to create an equivalent equation for the following: n + 6 = 10 Looking at the question (and thinking about "opposite operation) the logical thing to do would be to use subtraction. n + 6 - 6 = 10 - 6 n = 4 "n = 4" is equivalent to "n + 6 = 10". Sometimes the question will ask you to use all four operations (adding, subtracting, multiplying and dividing). Some questions lend themselves better than others to a particular operation. 2n + 4 = 10 (use subtraction...- 4) n - 7 = 8 (use addition...+ 7) 6n = 24 (use division...divide 6) n = 9 (use multiplication...pick a number...or addition...or subtraction) Patterns and Relations We begin this unit using an "input/output" machine. There is a list of "input" numbers (ones that you feed into the machine) and an operation (or operations) that you complete using the input numbers, resulting in the answers (output numbers). +4 input output 1 __ 2 __ 3 __ 4 __ The operation at the top is +4. You take the first number in the list of input numbers (1) and add 4 and get the output of 5. Then you take the second number (2) and add 4 to get 6...and so on. The completed chart looks like this: +4 input output 1 5 2 6 3 7 4 8 Sometimes the rule at the top asks you to do more than one operation. x 2 - 3 input output 2 1 4 5 6 9 8 13 The operations must be carried out as they appear in the rule. There are "pattern rules" we must know. The pattern rule for the input numbers refers to what is happening to the numbers in the input row. For example, in the first example the rule is "starts at 1 and goes up by 1 each time". The pattern rule for the input in the second example is "starts at 2 and goes up by 2 each time". There is also a pattern rule for the output numbers. In the first example it is "starts at 5 and goes up by 1 each time". The second output pattern rule is "starts at 1 and goes up by 4 each time". The last pattern rule involves the input "in relation" to the output. In other words, what did we do to the input numbers to get to the output numbers? In the first example it was "add 4". In the second one it was "multiply by 4 and subtract 3". Finding the Pattern Rule Which Relates the Input to the Output When you are given a partial or complete chart (or table) you may be required to find the three pattern rules. The pattern rule for the input and the pattern rule for the output if usually fairly simple. The pattern rule that relates the input to the output is a bit tougher because it involves figuring out what was done to the numbers on the left to get the numbers on the right. Whatever the rule is, it must be true, or work, every time. Example: _________ input output 1 7 2 8 3 9 4 10 Start by looking at what is happening to the numbers as they go from input to output. For example, in this table of values the first input number goes from 1 to 7. The second one goes from 2 to 8. In each case the value goes up. We can assume that the operation is likely adding or multiplying (because these cause the value to rise). First we believe it is a one-step rule so we start by asking ourselves what can we add to 1 to get to 7? The answer is 6, so the rule COULD be input + 6. Try it on a few other numbers to see if it works. 2 + 6 = 8, 3 + 6 = 9...seems to work. The pattern rule therefore is "input + 6". If that did not work we would try multiplying. What times 1 equals 7? 1 x 7 = 7. See if it would work on another input number. 2 x 7 = 14 (the output for the second number is 8, not 14 so that proves that "input x 7" is not the correct answer. Helpful Hints Often there are hints on what the pattern is for the input in relation to the output (what the machine is asking you to do to the input number to get the output number). Always begin thinking that it is a one step pattern rule. If the numbers go up from the input to the output, try adding or multiplying the input by a number to see if it works. Remember, it has to work for the whole list of input numbers. If the numbers go down, then try division or subtraction. If the one step attempts don't work then you know the pattern rule involves two steps. If the numbers go up chances are your first step is multiplication. If the numbers go down your first step is likely division. Solving the Rule Always assume that the pattern rule is one step to begin with. As mentioned above, if the numbers rise try an adding operation (it must work all the way down the chart). If that doesn't work try a multiplication. If neither work, then it is likely a two step rule. Most two step rules begin with multiplying, so that would be a good place to start. Look at the input numbers. If they go up by one each time (the input pattern rule), then look to see what the output numbers go up by. Whatever it is, that is your multiplier. Start with that. example: input (n) output 2 7 3 10 4 13 5 16 In the chart above, the input numbers go up by one. The output numbers go up by 3. That is a hint that 3 should be your multiplier. Begin your pattern rule by multiplying by 3. 2 x 3 = 6. Now, the first answer on the output side is 7 (not 6). How can we go from 6 to 7? By adding 1. So the pattern rule that relates the input to the output COULD be "input x 3 plus 1". Try it on the second number to see if it comes true. 3 x 3 + 1 = 10. Bingo! Nailed it! The same pattern rule should work all the way down the row of input numbers. Try the same for numbers that go down. Start with one step subtraction or division. If those do not work then the rule must be two step. As with multiplication, if the numbers go down, and it is a two step rule, then the first step is likely division. Look at all the input numbers. Pick something that will divide evenly into all of them. example: input (n) output 4 5 6 6 8 7 10 8 In the above example, 2 divides into all the input numbers evenly. Start by dividing by 2. Take the first input number (4). 4 divided by 2 = 2. Now, how do we get to 5 (which is the first output number)? By adding 3. So the rule COULD be "input divided by 2, then add 3". Try it for the second input number. 6 divided by 2 + 3 = 6. It works. It should work all the way down the line of input numbers. Coordinate Graphing Coordinates are listed in parentheses and are represented by 2 numbers (2,3). We use these numbers to find a point on a grid. The first number goes to the right and the second number goes up. Begin at zero on the grid (lower left hand corner) and go to the right the number of jumps given in the coordinates. For example, if the coordinates are (4,6), go to the right 4 places. From THAT point, go up 6. Mark that spot. You always begin your search for the point from the zero point. If a zero ends up in your coordinates, then you do not move in that direction. For example, if the coordinates are (3,0), you go to the right 3 and then up zero (don't go up). If the coordinates are (0.6), you go to the right zero (stay put) and then up 6. Where do the coordinates come from? The coordinate pairs come from the table of values (chart) that is created. n + 4 input (n) output 1 5 (1,5) 2 6 (2,6) 3 7 (3,7) 4 8 (4,8) Take these coordinates and plot them on the grid (graph). Students should be able to go from the chart to the graph but also be able to create the chart FROM the graph. Order of Operations For questions that have multiple operations (adding, subtracting, multiplying and dividing) in them, there is a certain order that has to be followed to reach the correct answer. The order is: 1. Parentheses (do all operations in "brackets" first) 2. Exponents (these will come in grade 7) 3. Multiplication and Division (starting from the left do these as they appear) 4. Adding and subtracting (starting from the left do these as they appear) Example: 4 + 7 x 2 + (3 - 2) + 8 x 2 Answer: (Shown one step at a time) 4 + 7 x 2 + 1 + 8 x 2 4 + 14 + 1 + 16 18 + 1 + 16 19 + 16 35 If you have a parenthesis that has a multiplication/division and an adding/subtraction, you must do the multiplication/division first. Example: 3 + (2 + 6 x 8) 3 + (2 + 48) 3 + 50 53 Each time you solve a portion of the equation, begin a the left again. Do not try to backtrack. Example: 43 - (7 x 2) 43 - 14 Do the (7 x 2) first like you are supposed to =14. Then go back to the first of the question and answer 43 - 14 NOT 14 - 43. Integers These are numbers than can be both positive (+) or negative (-). All numbers we have dealt with so far have been positive numbers. If there is no sign in front of the number it is assumed to be a positive number. positive numbers: all numbers greater than zero in value. negative numbers: all numbers less than zero in value. When placing integers on a number line, the zero will be in the middle of the line and to the right would be positives (+1, +2, +3, etc.) and to the left would be negatives (-1, -2, -3, etc.). The farther to the right you go the higher the "value" of the number will be and the farther to the left, the lower the value. Therefore, a -23 is lower in value than a + 6 despite the fact that it is a "23". Remember, it is a negative 23 so, much smaller than zero, where +6 is above zero. The most common areas where you would find integers are in temperature (where measures can be above and below zero) and in games like golf (where below par is considered a negative). You should be able to recognize the vocabulary of positive and negative phrases. Positive examples: more, gain, add, increase, over, savings, deposit, after, up, above, in, to the right. Negative examples: less, loss, subtract, decrease, under, debt, withdrawal, before, down, below, out, to the left. Question: Which integer is suggested here: Maria took two steps backward. Answer: -2 The submarine was at 20 metres below sea level. Answer: -20 Bob made 30 dollars mowing the lawn. Answer: +30 Comparing and Ordering This involves placing integers on a number line and telling which integer is larger or smaller that the others. Remember, all negatives are smaller than positives, regardless of the numbers. For example, - 34 is a lot smaller than + 2, despite the fact that 34 is bigger than 2. It is a NEGATIVE 34, therefore a long ways BELOW zero. Per cent Per cent is another way to represent a part of a whole, like fractions, decimals and ratios. The words "per cent" literally means "out of 100". If you have a fraction out of 100 then it is easy to figure out the per cent. Example: 34 = 34% 87 = 87% 100 100 There are certain benchmark per cents you should know. 50% is half of something, 25% is half of that and 75% is 3/4 of something. You can use these benchmarks to estimate other per cents. If you know what 25% looks like, then 30% wouldn't be that far away. Per cent to Decimal and Back The fraction 48 can be read as forty-eight hundredths, which is written 0.48. 100 The fraction 48 is the same as 48%, so 48% is also equal to 0.48. 100 Examples: 37% = 37 = 0.37 72% = 72 = 0.72 100 100 Finding Per cent What do we do when the fraction is NOT out of 100? If you can, turn the fraction into an equivalent fraction out of 100. We have done this before with fractions and ratios. Example: 20 x4 = 80 25 x4 = 100 The fraction was out of 25, but 25 goes into 100 four times. Multiply the 25 by 4 and the 20 by 4 (remember, what you do to one term you have to do to the other to keep it equal in VALUE). Now you know that 20/25 is equal in value to 80/100, so 20/25 is equal to 80%. What about 7 ? 7 x 10 = 70 10 10 x 10 = 100 Multiply both terms by 10 (because 10 goes into 100 ten times). 7/10 equals 70/100, or 70%. What do we do if the fraction cannot be changed into a fraction out of 100? We divide the bottom number into the top. That will give us a decimal. We already know how to change a decimal to a percent (see above). Example: 3 8 Use the fraction sign like a division sign (3 divided by 8, or how many times will 8 go into 3?) You may think this is impossible. Remember, we can add zeros to the 3 after we place the decimal (3.0000000). Divide as if the decimal isn't there and then place the decimal afterward. The answer will be zero point something (o.___). Stop dividing after 3 decimal places and round off your answer to two decimal places. Then change the decimal to a percentage. For the example below, remember I can't make a division sign. Imagine it there. 0 .375 8 3.000 0.375 rounds off to 0.38, or 38%. 24 60 56 40 40 0 There will be word problems where you will have to figure out the fraction from the information, and then create the percent. Remember to show all your work no matter how easy the question. Example: John made 21 out of 50 on his math test. What was his per cent? fraction: 21 x2 = 42 = 42% Answer with a sentence. John made 42% on his math test. 50 x2 = 100 Fractions Fractions are a way to represent parts of a whole. The bottom number is the denominator and tells the total parts. The top number is the numerator, or the pieces of the whole. Example: OOOOXXXXX There are 9 objects in the above diagram. That would be the denominator in the fraction. There are 4 circles. The fraction of circles to whole shapes would be 4/9 (four ninths). The fraction of Xs to whole shapes would be 5/9 (five ninths). The two numerators should add up to the total number of shapes: 4/9 + 5/9 = 9/9 (or the "whole" diagram). Equivalent Fractions Sometimes fractions have different terms (numerators and denominators) but have the same "value". These are called equivalent fractions, similar to one loonie being equal in value to four quarters or 10 dimes. Using the above diagram as an example, if we double all the shapes we end up with 8 circles, 10 Xs and 18 total shapes. OOOOOOOOXXXXXXXXXX The fraction of circles to total shapes is now 8/18. 4/9 is equal in value to 8/18. We can check this by multiplying the terms in the first fraction by 2 (both numerator and denominator) to see if we get the numbers in the second fraction. 4 x2 8 9 x2 18 Even though the numbers are different, the proportion is the same. We can create equivalent fractions by multiplying or dividing each term in the fraction by the same number. What you do to one term you MUST do to the other to keep the same proportion and value. Examples: 1 x2 2 2 x4 8 18 divide 6 3 2 x2 4 3 x4 12 24 divide 6 4 1/2 is equivalent to 2/4. 2/3 is equivalent to 8/12. 18/24 is equivalent to 3/4. You MUST remember to do the same to the top number as you do to the bottom number to create equivalent fractions. Improper Fractions Sometimes you get fractions that have a larger top number (numerator) than the bottom number (denominator). These are called improper fractions. They are called that because we are used to seeing the bigger number on the bottom. These fractions have a value greater than one. Example: OOOOOO OOOOXX The first diagram has 6/6 circles compared to the total shapes. The second diagram has 4/6 circles compared to total shapes. Altogether there are 6/6 + 4/6 = 10/6 circles to total shapes. We have 1 whole shape with circles and 4/6 of another shape, so that gives us 1 4/6 (one and four sixths), or what we call a mixed number (one with a whole number and a fraction). Therefore, 10/6 is equal to 1 4/6. We can move from improper fraction to mixed number (or whole number) and back by using a number line, drawing pictures or using a shortcut. Shortcut method: 12 5 The fraction line is a division line, so the fraction can be read as 12 divided by 5. If we do the math we end up with 5 going into 12 twice with 2 left over. The whole number is 2 and the fraction is 2/5 (using the same denominator in the original fraction). 12/5 = 2 2/5 (two and two fifths). Let's try another. 27 4 27 divided by 4. The answer is 6 remainder 3. So the mixed number would be 6 3/4 (six and three fourths). Sometimes when you divide there is no remainder. That means the improper fraction equals a whole number. Example: 35 7 35 divided by 7 = 5 with no remainder. 35/7 = 5 To go the other way...begin with the mixed number. example: 2 2 (two and two thirds) 3 Start by multiplying the whole number (2) by the denominator (3). The answer is 6. Then add the numerator (2) and get 8. The 8 becomes your numerator for your improper fraction and the denominator is the 3 (same as it was). Answer: 8/3 2 2/3 = 8/3 Try 4 3 5 Multiply the 4 by the 5, then add the 3= 23. That's the top number. The 5 is still the bottom number. 4 3/5 = 23/5 Ratios Ratios are comparisons between objects (using numbers). For example, if there are 12 boys in the class and 14 girls, the ratio of boys to girls is 12 to 14. The ratio of girls to boys is 14 to 12. The ratio of boys to the total would be 12 to 26, and girls to total, 14 to 26. The order is important. Pay attention to what is being asked and make sure the order is the same as is being asked. Example: OOOOXXXXX What is the ratio of circles to Xs in this diagram? The ratio of circles to Xs is 4 to 5. What is the ratio of circles to total shapes? The ratio of circles to total is 4 to 9. There are two main ways to write a ratio. 1. Using the word "to" (4 to 18). 2. Using a colon (4:18). We still use the word "to" when saying the ratio out loud. There is a third way but it should only be used when you are comparing the part of something to the whole thing. We did that when we compared the circles to the total shapes (4 to 9). We can write this as a fraction (4/9) because a fraction IS a ratio comparing the part to the whole. Ratios can have more than two terms. You have 3 red marbles, 4 green marbles and 7 blue marbles. The ratio of red to green to blue marbles is 3:4:7. Equivalent Ratios We create equivalent ratios the same way we create equivalent fractions, by multiplying or dividing each term by the same thing, to keep the proportion the same. We can use this to figure out ratio word problems. Example: The ratio of girls to boys in an average class is 12 to 13. If we had three classes, how many boys and girls would there be (assuming the proportion is the same)? 12 to 13 (multiply each term by 3 to represent the 3 classes) x3 x3 36 to 39 There would be 36 girls and 39 boys. The ratio of eggs to cups of milk in a recipe is 2 to 1. If we wanted to use up all the eggs in the fridge (6), how much milk would we need for the recipe? 2 to 1 x3 x3 6 to ___ We went from 2 to 6 (eggs) by multiplying the 2 by 3. We have to do the same to the 1. 1 x 3 = 3. We would need 3 cups of milk. Multiplication Methods 1. Traditional 142 Place the larger number on top. x 38 Always make a quick estimate so you’ll know if your answer is close. Multiply the 8 by all the digits in the top number (one at a time). 8 x 2 = 16. Place the 6 in the ones place and carry the 1 over the 4. 8 x 4 = 32, then add the 1 for 33. Place a 3 and carry a 3 over the 1. 8 x 1 = 8, then add 3 for 11. The answer for the first row is 1136. 31 142 X38 1136 Move over to the 3 in the ten’s place and multiply it by each of the top numbers (one at a time). Place a zero in your answer to represent the move to the ten’s place. 3 x 2 = 6. Put a 6 next to the zero. 3 x 4 = 12. Put a 2 next to the 6 and carry the 1 over the 1. 3 x 1 = 3, plus 1 is 4. Put the 4 next to the 2. The second line is 4260. 1 142 X38 1136 4260 Now add the two lines together to get the final answer. 142 X38 1136 4260 5396 2. Break it Down Break down each number into it’s parts and multiply them individually. 42 8 x 2 = 16 x 38 8 x 40 = 320 30 x 2 = 60 30 x 40 = 1200 1596 3. Bow Tie This is really just another way to use Break it Down. You draw a line from one number to the next as you multiple them, creating a "bow tie" shape between the numbers. It helps to make sure all the numbers are multiplied. This only works with two-digit by two-digit multiplication. 4 2 X 3 8 8x2= 16 Make sure your place Add 30x40=1200 values are lined up. 40x8= 320 30x2= 60 1596 4. Area Model Again, another method of breaking down the question. The formula for area of a rectangle is A=l x w (length times width). Each number is broken down and these numbers are used for the length and width of the rectangles. See below. (This site will not allow me to create lines to form rectangles so, perhaps, you can imagine them.) One rectangle is 40 by 30, another is 30 by 2, a third is 40 by 8 and the last one is 8 by 2. 40 2 30 40x30=1200 30x2=60 8 40x8=320 8x2=16 Now, add the totals of each rectangle for the final answer. 1200+60+320+16= 1596 Notice that the equations are the same as the "Break it Down" and "Bow Tie" method. Students have been given a sample of each method and have placed it in their math homework book. Decimals The place values on the decimal side are very similar sounding to the ones on the whole number side. Remember the whole number ones? ones, tens, hundreds, thousands, 10 thousands, 100 thousands, millions, etc. On the decimal side they are treated like fractions. There is no ones on the decimal side so we start with tenths. The "th" shows that the number is less than one, a fraction of 10. If there was a 3 in the first decimal place (for example, 0.3) that would represent three tenths, or 3/10. If the number was 0.34, it would represent three tenths and four hundredths (or we could say thirty four hundredths (34/100). You will have to know how to say the number properly and show the number in expanded form. For example, 2.875 would be two and eight hundred seventy five thousandths ( 2 and 875/1000). It can also be written as two and eight tenths, seven hundredths, five thousandths. The expanded form would look like this: 2 + 0.8 + 0.07 + 0.005. When comparing and ordering decimals, start at the left hand side of the numbers and compare each digit. Go until you find one that is larger or smaller and that will tell you which number has more value. The number of digits in the number does not matter. For example: Which is greater (has more value)? 4. 05289 or 4.07 At first glance it appears that 4.05289 is larger because it has more digits. Look closely at each digit starting from the left. Both have 4 ones, both have 0 tenths, the first one has 5 hundredths while the other one has 7 hundredths. We do not need to look any further. It doesn't matter that the first number has extra digits because 5 hundredths is less than 7 hundredths. 4.05289 is less than 4.07. When trying to find a number that fits between two decimals (and there appears to be no room) simply add a zero to each and take a another look. Find a number between 4.45 and 4.46. There doesn't appear to be one. If we add a zero to each decimal (4.450 and 4.460) suddenly there appears to be at least 9 numbers to choose from: 4.451, 4.452, 4.453, 4.454, 4.455, 4.456, 4.457, 4.458, 4.459. By adding the zero at the END of the decimal you are NOT changing the VALUE of the number so it is okay to do it. Operations with Decimals When adding and subtracting decimals, make sure all place values are lined up, including the decimals. Add or subtract normally and then move the decimal straight down into your answer. example: 345.67 + 23.22 368.89 When multiplying, there are two ways to place the decimal. First, multiply the numbers as if the decimals in the question do not exist. When you get your answer, either: 1. Count the decimal places in the question (number of digits to the right of the decimal in all the numbers you are multiplying). Then starting from the right (in your answer) move the decimal that many places to the left. For example: 2.13 There are two decimal places in the first number (1 and 3) and x 2.4 one decimal place in the second number (4). That makes 3 5112 decimal places in total. Start at the right of the 2 and jump to the left 3 places, which would place the decimal between the 5 and the 1. The answer is 5.112. 2. Estimate: In the above question you are multiplying 2 and a little by 2 and a little (the decimal amounts are very small). 2 x 2 = 4. Your answer should be around 4. The digits in the answer are 5112. Where would you place the decimal to make the answer closest to 4? 5112? 511.2? 51.12? 5.112? or .5112? Obviously, the one closest to 4 would be 5.112 (5 and a little bit). When dividing, divide as if the decimal does not exist and place it after you get your answer. The traditional long and short division methods will work better for decimal division due to the fact that it is easier to take care of remainders (which are converted to decimal). If using long or short division and the answer works out evenly, simply mover the decimal in the question straight up into your answer. If your answer digits are placed properly then the task will be easy. Let's do an example: (try to imagine the division sign) 2 3 8 2 4 7.6 4 0 7 6 1 6 1 6 0 The 2, 3 and 8 are placed in the proper places so move the decimal in the question (between the 7 and 6) straight up into your answer. The answer is 23.8. If the question does not work out evenly, you need to add zeros to the number inside the division sign and continue to bring numbers down and divide until it does (or for two more decimals). Some will never work out evenly so two decimal places will be far enough. Let's try another example: 1 8 9 3 3 (3 repeats) 3 5.6 8 0 0 3 The question does not work out evenly so 2 6 we add a zero to the question (after the 8) 2 4 and continue to divide. Then another. As you 2 8 can see, it will never work out evenly so we 2 7 stop adding zeros after placing 2 of them. 1 0 9 1 0 9 1 Move the decimal straight up into your answer (between the 1 and the 8). The answer is 1.89333333...This program will not allow me to do it, but we show a repeating decimal by placing a straight line __ over the the digit that repeats (in this case the 3). If you are using super seven or your digits in your answer are not properly placed you can figure out where the decimal goes by estimating. For example, the first question (47.6 divided by 2), we can eliminate the decimals and/round off and divide in our had 48 divided by 2 = 24. So the answer should be around 24. What were our options? 238? 23.8? 2.38? .238? Which makes the most sense? In our second question, we estimate 6 divided by 3 = 2. The answer should be close to 2. What are our options? 1893? 189.3? 18.93? 1.893? .1893? Which makes the most sense this time? Which is closest to the VALUE of 2? 1.893. Lang Arts Notes Students have been asked to attempt to read 40 books during the school year. Many have had a great start but some have fallen behind. Ask your child where they are on this journey. Some may need a push. |