What type of math activity do you most like to have your middle school math students work on?
For me, it's almost always been problem solving. This could include word problems that apply specific math concepts, word problems that incorporate a variety of math concepts, logic puzzles, or word problems that focus on problem solving strategies (create a table, make an organized list, find a pattern, work backwards, draw a picture, etc). I love using problems that have more than one correct solution, so students can share the thinking that leads to different answers.
When we work on problem solving activities, I often have students work together, so they can model for each other and share/listen to each others' thinking and reasoning.
I wrote the "Party Planning" problem to give students practice with decimal operations and with solving problems with multiple solutions. To solve the problem, students worked with one or two partners to come up with combinations of foods that Reggie could buy for a party. To find their solutions, students needed to add decimals; multiply if they were going to include several of one item; and possibly subtract, if their total was over $50.
Student Conversation and Feedback
I loved listening to the kids' conversations as they worked on this problem. I heard comments like, "No one eats pretzels," or, "I'd choose candy and chips over pretzels," and so on.
The students had a few important questions for me, as they were pretty serious about this planning.
"Is this a "regular" party or like a sleep-over party, because the kind of food would depend on how long the party is."
"How big is the container of ice cream?"
"How big is the bag of candy?"
Could you use a quick math activity to help your students practice identifying decimals in standard and word form? How about some comparing and ordering of decimals? I've got an activity that covers all of those for you:-)
When I created this one, we were just beginning our work with decimals (in grade 6), and my students had done a little bit of work with writing decimal numbers in word form. They had also worked on comparing decimals several times during the year, in our Daily Warm Ups book.
Using the Decimal Matching Activity
The first step in the activity is to match each card with a decimal number in standard form to the card with the correct word form. I allowed students to work alone or with one partner, and the matching didn't really take that long. I did have similar numbers (like 9.68, 9.068, 9.0068 etc), so that the students had to read carefully and take some time to compare those similar numbers.
Free Number Puzzle!
Turning decimals into a fun, Sudoku-like puzzle is a great way to help students work their way through the different decimal operations.
I love solving Sudoku puzzles using whole numbers, and I encourage students to work on them as well. But when I replace the whole number with decimal problems, I’m able to create a logic puzzle that also gives students a new way to practice skills that can be a challenge to master.
Sudoku puzzles are fun and interesting. They require students to problem solve, in order to ensure that every row, column and group of squares only has one of each number. Some students develop strategies when approaching a puzzle; some learn to use guess and check quite often:-)
When working with decimals in a Sudoku puzzle, students need to consider their target numbers, as well as how to complete the necessary operation. This makes decimal practice a bit more interesting and engaging than working on one problem after another on a worksheet.
As with whole number Sudoku, I start decimal Sudoku with a few squares filled in, so students have a starting point. (If students have never tried Sudoku puzzles before, I recommend starting with a regular Sudoku puzzle, to teach students how they work.)
So far, I’ve only used a 4 X 4 grid, which makes figuring out the target numbers fairly easy. All of the squares have a decimal problem with a missing addend in them (in this example). Within each 2 x 2 section, there is one completed problem, with the target number of 1, 2, 3, or 4 already filled in. The object is for students to figure out the target number for each square and then find the missing decimal number in each individual square. Every row and column must each contain 1, 2, 3 and 4 as the answers to the decimal problems (these are the target numbers). Every 2 x 2 section must also contain 1, 2, 3 and 4.
For example, on the answer key shown here, you can see the target numbers of 1-4 in each row, column and section, and you can see the completed decimal problems. Everything in black (target numbers and decimal numbers) is given. Everything in green is what the students must find (target numbers and missing decimal addends):
I've only used the addition problems (which actually require them to subtract:-), but I plan to try the other operations as well and create larger puzzles (6 x 6 and maybe 9 x 9).
To make students really think things through, I may mix up the operations! For instance, in a 4 X 4 puzzle, the first column could be two multiplication problems and two addition problems. This should keep students paying close attention.
These could be great to complete on mini dry erase boards - easy to erase any guesses that don't work out!
I'm really liking the math wheel idea, so I created a new wheel for fraction, decimal, percent conversions:-)
How to use this resource (this information is also in the free download):
Around the outside of the wheel are the different conversion headings – you can use the wheel to introduce the conversions, filling in just the ones you are covering each day. Or, you can use it to review all the conversions at once. In either case, the wheel can be kept in students’ notebooks as a reference/study tool.
1) I like to begin with decimal to percent and percent to decimal. In the arrows in these sections, you’ll see x 100 and ÷ 100. It think it’s important that students understand that these are the operations being used for these conversions before giving them a shortcut, so I let them use calculators to complete the examples. Once the examples are complete, I ask the students to look for the pattern – what happens to the decimal point in each of these cases?
We decide on the “shortcut” rules together and then write them at the bottom of those sections.
2) The fraction to percent and fraction to decimal sections have the rules written already, so the examples just need to be completed.
I always relate fraction to percent to students grades. By the time we get to this topic during the year, students have been figuring out their grades for months (I never write their percentages on their assessments – they need to calculate
them). They know how to find their percentage if their quiz grade was 6/8 or their test was 48/52. However, sometimes they need a reminder that this official fraction to percent “rule” is the same thing they’ve been doing for months! I have them
write a little reminder in that section - “just like test grades!”
3) For percent to fraction, students need to remember that percent means “out of 100,” so the percent number will always go over 100. Then they must reduce.
4) I find that decimal to fraction is sometimes tricky for students. When they have trouble, I ask them to read the decimal number according to place value (“How do you say this number, using tenths, hundredths, or thousandths, etc.?”). Once
they speak it, they know how to write the fraction – 0.27 is 27 hundredths, which is 27/100. After completing the examples, we discuss the idea that the denominator will be whatever the last decimal place is (10, 100, 1000, etc.) and the numerator will be the digits in the decimal number. We write this rule as simply as possible.
5) Students then complete the 10 problems around the page. Above each number is the conversion to complete (F to P, P to D, etc.) They can then color the rest of the wheel background.
I had a great time coloring my answer key! These could make a fun decoration as well:-)
I hope you can use it!!
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Do your middle school math students like to play math games? Mine do, but over the past few years I've noticed that many of them aren't familiar with some of the games I played when I was a kid, like Yahtzee, for example. So, as we started working on converting fractions and decimals, I decided to create a game to make practicing the conversions more fun AND give them some more game experience! I based it on the idea of Yahtzee:-)
Here's how it works:
Students roll four dice, and pair the dice up to create "target numbers" that are either decimals or whole numbers.
For example, a student rolls 1, 2, 4, and 6. From these dice, the student may create any two of the following decimal (or whole) numbers:
½ = 0.5 4/1 = 4
¼ = 0.25 4/2 = 2
1/6 = 0.1666... 4/6 = 0.666...
2/1 = 2 6/1 = 6
2/4 = 0.5 6/2 = 3
2/6 = 0.333... 6/4 = 1.5
Once a player has chosen two target numbers, he or she finds the score by adding the dice that were used for each decimal (or whole number). If the player chose to use 1 and 4 to get 0.25, he or she adds 1 + 4 for a sum of 5 to place in the score column. If the second choice used 2 and 6, to equal either 0.333...or 3, then sum of 8 would go in the appropriate column as the score.
On the next roll, this student rolls 1, 1, 3, and 5. This student can pair 1 and 1, to get 1, and pair 3 and 5 to get either 0.6 or 1.666... The score for 1 is 2 (1 + 1) and the score for 0.6 (or 1.666) is
8 (3 + 5).
In many cases, students' scores will be the same, but some of the decimals can be found with different combinations of numbers (1 and 3 = 0.333..., and so do
2 and 6, so students could have a score of either 4 or 8). Some students will notice this sum difference and go for the combination that will give them the higher score....bringing in the possibility of using some strategy, for those higher level thinkers.
The students have really enjoyed playing this game. They do need a few examples at the start, to understand exactly how the game works, so if you decide to try the game, be prepared to go through a few turns together.
You can create a score sheet like this on your own, or go to TPT and use what I've created. Detailed instructions are included, and a complete answer key of highest and lowest possible scores for each target number are included as well. This is handy to quickly check student score cards as you check in on their games.
If you give it a try, please let me know how it goes!
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Math rules. How often have you found that students are taught "tricks" to remember math rules? How often do they make procedural mistakes even though they've "learned" the rules?
I have taught decimal operations for more than 20 years, and I have seen, time and again, students who know how to add and multiply decimals but then follow the wrong "rule" for the operation they are completing. Line up decimal points when adding or when multiplying? "Jump" the decimal point over when adding and subtracting? Or is that multiplying? They don't remember when to use which method to place the decimal point.
Students' Thinking About Decimal Operations
So, this year, as we approach the decimal unit, I've been feeling like I don't want to talk about the rules for where/how to put the decimal point. I want to focus on logic. Today that feeling was reinforced when I asked my students to solve 35.2 + 7.489 and then explain why their answer made sense. Here are some of the answers and reasons (I didn't teach this yet, but they learned it last year):
"0.11009 makes sense because I tried my best and if I remember correctly, addition problems you don't need to line the decimals together"
"0.7838 makes sense because when I added I knew that it doesn't matter how it's lined up"
"78.42 - I added 9 and 2, then 1, 8 and 5. Next I added 1, 3, and 7. Finally I added 7 and 0 and I put the decimal in the middle."
"7.841 makes sense because with adding you only have to add the decimals on the top. Then you add and finally add the decimal back in."
"426.89 because I put the decimal point four spaces back because there are four numbers behind it"
"79.41 makes sense because you do it just like an addition problem (that's how I remember it anyway)"
"7.841 makes sense because you add like normal and take the decimal from the farthest out and put it with the answer"
A few correct answers, with reasons:
"42.689 - this makes sense to me because this is how I learned it. You do simple addition, but line up the decimal points"
"42.689 makes sense because I used what my fifth grade teacher taught me, line up decimals, add zeros so everything is lined up and then solve."
"42.689 - I don't know how it makes sense, but it's how I learned to do it."
Of the 120 students in my classes, only 8 said the answer made sense because "35 + 7 is 42" or because "I estimated" or "when we're doing addition, we know we end up with a bigger number."
Now, that doesn't mean that they didn't think about those things, but to them answers seemed to "make sense" when they followed the rules - even if the rules are remembered incorrectly; students got right and wrong answers and they all made sense because that's "how they learned it."
So, what is the point of teaching rules? Especially to those students who are a little weaker in math - if they can't remember the right rule, they can't tell if their answer is reasonable! They need to develop their number sense.
In the past, I have asked students to estimate the answer first, so they know if their answer is reasonable, and I have required them write these estimates on their tests. But we've also talked about the rules. I'm thinking that if I take the focus off the rules and put extra focus on the estimating/reasonable answer idea, students will be better able to identify reasonable answers and will feel less dependent on the rules.
Multiplication and Division of Decimals
I know that multiplication and division logic will be more difficult. Problems like 23.5 x 4.428, won't be as bad because there are whole numbers involved. This could be estimated as 25 x 4 = 100. So when placing the decimal point in 104058, students should place it so that the answer is about 100 - not 10, or 1, or 1000.
Multiplying 23.5 and 0.7 may be more confusing, but this will be the time to help them understand why the answer should be smaller than 23.5....but more than half of 23, since 0.7 is more than 0.5.
I think division will be the most challenging, as far as determining reasonable answers, and I need to think about this one a bit more. However, we have already done this activity I found on YouCubed -"Too Big or too Small Maze Board." In attempting to create the largest number possible (using a calculator to compute), many students have already made the discovery that dividing by a number less than one gave them a larger number, while multiplying by a number less than one gave them a smaller number. No rules were taught - they found this "secret" on their own. This will be great to reference and discuss when we begin working on the multiplication and division of decimals.
We'll see how it goes!
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