I think the explanations are pretty clear here in the picture. The top relates to yesterday's explanation. The only thing I did not cover in the picture was absolute value. Absolute value is basically asking a number how far it is from zero. It is never a negative answer to that question. So the absolute value of 3 is 3. The absolute value of -3 is also 3. I3I=3 AND I-3I=3
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Ratios, they seem to be a problem. The students are having trouble connecting ratios to their equivalents. The ratio 24:36 can be reduce (very similar to reducing fractions), by dividing each side by the same number. 24:36, if I divide each side by 2 equals 12:18. I can divide that again by 2, which gives me 6:9. Now I have to divide by 3 and get 2:3. So the ratio of 24:36 can also be shown as 2:3.
When we added in rate tables and unit rates, the confusion worsened. A rate table simply puts ratios in the form of a table. Bob's number of books 24 12 4 2 Jill's number of books 36 18 6 3 (Sorry there are no lines around it I'm at home and this computer doesn't cooperate with uploading pictures.) But whatever the ratio is for the first column in the rate table, it stays consistent throughout the table. If Bob had 8 books, Jill would have 12. If Bob had 5 books, Jill would have 7.5 (I know you can't have .5 of a book, but 2x2.5=5, so 3x2.5=7.) The ratio must remain constant. Then we tackled unit rates. All this means is one of the numbers in the ratio or rate table is one. The problem usually tells which number should be one. With Bob and Jill and their 2:3 ratio of books, the unit rate for Bob is 1:1.5. Jill's unit rate would be 1:2/3. I hope this helps. I'll try to get something posted sooner for our next adventure...absolute value! It is all about sharing (fair shares among brothers and sisters really IS important). They practiced sharing gummy worms with differing amounts of color/flavor segments in different ways. A2 demonstrates two distinct interpretations of one kind of sharing (but with the same result in how much of a segment of flavor or how many segments of flavor each person gets). The top one in A2 shows sharing 6 whole flavor segments, then dividing the remaining two flavor segments into 6 smaller but equal sections. Each person would get one whole flavor segment and 1 /3 of another segment (1 1/3 segments). The second one in A2 shows dividing each segment into thirds, and then doling those out evenly. Each person would get 1/3+1/3+1/3+1/3 or 11/3 segments.
Tomorrow, we throw a monkey wrench into the plan and make it NOT FAIR SHARES. I can hear the siblings complaining now! This is my giant version of the fraction strips students made in class last week. We are showing (visually) equivalent fractions. 5th grade math lays a strong foundation with fractions, now we build on that foundation. Just as with our last unit, a strong background in multiplication and division facts really helps.
This week will see the lessons require flexible thinking about fractions, figuring out how to share things divided by one fractional unit into other fractional units, and linking rate charts and ratios to the fractional foundations in place. A couple of important lessons from Tuesday's test: many still struggle with Order of Operations, many still confuse when/how to use Venn diagrams to find GCF and LCM, many confuse when to use common factors and when to use common multiples, and careless errors can bring down math grades. Order of Operations P-parenthesis E-exponents MD-multiplication and division AS-addition and subtraction Math problems must be worked in the order of these rules. Multiplication and Division are of equal importance, so are worked left to right. Addition and Subtraction are of equal importance, so are worked left to right. I am working with my students to not skip steps. Examples: I have an earlier post about using Venn diagrams to find GCF and LCM. The main problem I saw was people who attempted to use T-charts (or finding factors of two numbers) to create their Venn diagram. Using the Venn diagram method requires finding the Prime Factorization for two numbers and placing the shared prime factors in the middle of the Venn diagram, then placing the prime factors not shared in the circles that 'belong' to each separate number (see post from 9-4 for a more thorough explanation).
The problem of when to use common factors and when to use common multiples are addressed in the study materials (dated 9-13). The four pictures for today will review the basic concepts that will be on the very first test for the year (we've had quizzes, but not tests). Some of the problems will be very straight forward (like what is the Prime Factorization for...), but some will require the students to decide which math concept to apply. You can also look at my previous posts for additional explanations. 9-3 is how to find the prime factors of a number.
9-4 is how to use Prime Factorization to find GCF and LCM. 9-10 shows two ways to write expression (it demonstrates the Distributive Property). Other resources: They can access their Reflex math account via the internet at home (reflexmath.com). http://www.mathplayground.com/factortrees.html - This website gives excellent practice on finding the Prime Factorization of numbers AND using the Prime Factorization to find the GCF or LCM with a Venn diagram. If the pictures from today's post look like someone's kitchen counter, well that is MY kitchen counter--it doubled as a photography studio last night! I'll work on posting a better explanation later this week. I'll also post concepts to practice for the test next week.
More pictures will be posted as explanations, soon! The 6th graders are learning about the Distributive Property. This is the math property that shows things like 326 x 26 equals the same thing as 326 x 20 + 326 x 6. We just distribute the numbers around before doing the multiplication.
The picture shows a model they are currently practicing. It starts with the model for a math fact. I chose the number 15. 3 x 5 = 15. But we can break the 3 x 5 model up into two smaller sections 1) 3 x3 and 2) 3 x2. When you write this in a multiplication form it looks like (3 x 3) + (3 x 2) = 15. This form requires them to remember their order of operations (operations in parenthesis happen first). They can write the same kind of concept in a more compact form: 3 x (3 + 2). The 3 + 2 in the parenthesis being the two short lengths of the width of the model added together. It could also look like 3(3 +2). All of these get the same answer, 15. Why do they need this? It begins to lay the foundation (in very familiar territory) for future adventures in Algebra! For those still struggling with Prime Factorization, try this website: http://www.mathplayground.com/factortrees.html. It can help them practice Prime Factorization AND using prime factors to find the GCF and LCM. I have not played it myself, but it was recommended by another math teacher. First things first, I got the picture to appear upright!
We are still working on Prime Factorization and using the prime factors of two numbers to find GCF and LCM. I have shown one method of finding the prime factorization of both 30 and 75. This is the tree method. Once all the primes have been found, you can create the Prime Factorization. 30=2x3x5 and 75=3x5x5 Then, we create the Venn Diagram. Where the circles overlap in the middle, the prime numbers shared by the two numbers are placed; so with 30 and 75, 3 and 5 are shared and go in the center (or overlap) section. The 2 that is left from 30 belongs in the outer (non-shared) section of the 30 circle. The 5 that is left from the 75 belongs in the outer (non-shared) section of the 75 circle. If you multiply the two shared primes - 3x5 - you have the GCF of the two numbers. To find the LCM, all the factors in the Venn Diagram are multiplied. Example: 3x5 (the GCF) as well as the 2 that is in 30 section and the 5 that is in the 75 section or 2x3x5x5. This equals 150, which is the Least Common Multiple of 30 and 75. |
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September 2015
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