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A 5th grade standard that students seem to struggle to grasp would be "CCSS.MATH.CONTENT.5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols." Students always have a difficult time remembering the order of operations and incorporating them when they are writing their own expressions. I found an Illustrative Math task that I think does a great job of working to clear up any misconceptions associated with this standard. Students use the digits 1,2,3 and 4 along with all operations and parentheses to see how many different sums, quotients, differences and products they can make. This is Van de Walle approved because the "materials add depth to your information." It does this by showing students the versatility of the parentheses and operations. http://tasks.illustrativemathematics.org/content-standards/5/OA/A/1/tasks/1596

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A UDL assessment that could be paired with this would be writing an expression with parentheses and then writing a real world scenario to match. This assignment allows students to "illustrate through multiple media" in that they can express themselves with words and with numbers. It also tests their content knowledge because it shows they know what the operations and mathematical symbols mean.

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Fourth graders always seem to struggle with "CCSS.MATH.CONTENT.4.OA.B.4
Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite." I think students can find if a number is prime or composite but identifying all the factors for composite numbers is challenging if they don't know their multiplication facts. I found this task that students could pair an action with this task. Students realize that multiples of a number are like shutting that locker. This app would be approved by Van de Walle because of its authority. The site is "identifies the authors and their qualifications." (Van de Walle, )

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Alignments to Content Standards: 4.OA.B.4

Task

The 20 students in Mr. Wolf's 4th grade class are playing a game in a hallway that is lined with 20 lockers in a row.

The first student starts with the first locker and goes down the hallway and opens all the lockers.

The second student starts with the second locker and goes down the hallway and shuts every other locker.

The third student stops at every third locker and opens the locker if it is closed or closes the locker if it is open.

The fourth student stops at every fourth locker and opens the locker if it is closed or closes the locker if it is open.

This process continues until all 20 students in the class have passed through the hallway.

1. Which lockers are still open at the end of the game? Explain your reasoning.

2. Which lockers were touched by only two students? Explain your reasoning.

3. Which lockers were touched by only three students? Explain your reasoning.

4. Which lockers were touched the most?

IM Commentary

The purpose of this instructional task is for students to deepen their understanding of factors and multiples of whole numbers. This is a classic mathematical puzzle; often it is stated in terms of 100 lockers, so students have to make the connection to factors and multiples to solve it. In this version, students can just go through all the rounds of opening and closing locker doors and observe after the fact that there is a relationship between the factors a locker number has and whether it is open or closed at the end.

This task provides students with an excellent opportunity to engage in MP7, Look for and make use of structure (if they see early on that there is a relationship with factors and multiples) or MP8, Look for and express regularity in repeated reasoning (if they start to see and describe the pattern as they imagine students opening and closing the lockers).

Because the total number of lockers is only 20, students might answer the questions without thinking about the underlying reasons for their answers. In the first question for example, a student might say "1, 4, 9, 16 are all still open because I tried it out and those were the ones that were left open." If a student goes this route, the teacher can steer the conversation back to factors by asking the students if they notice anything special about the numbers of the lockers which are still open. Asking students to try a larger number of lockers (say 50 or 100) and repeating the game can also help students look for a pattern since going through all the rounds of the game becomes less and less feasible as the number of lockers increases. Once students see the pattern, they should be pressed to explain why it works out this way.

Solution

1. The lockers that are still open are perfect squares, specifically $1\times 1$, $2\times 2$, $3\times 3$, and $4\times 4$. To see why this is true, first note that:

The 1st student changes all the lockers with a locker number that is a multiple of 1.

The 2nd student changes all the lockers with a locker number that is a multiple of 2.

The 3rd student changes all the lockers with a locker number that is a multiple of 3.

The 4th student changes all the lockers with a locker number that is a multiple of 4.

and so on. So a locker is touched by a student if that student's number is a factor of that locker's number.

Now consider the factors of 16 which are 1, 2, 4, 8, and 16. This means the 16th locker was opened by the 1st student, then it was closed by the 2nd student, then it was opened by the 4th student, then it was closed by the second 8th student, and then opened one last time by the 16th student. Now note that the square numbers are the only numbers with an odd number of factors (all other numbers have factors which come in pairs.) Since an odd number of factors means the locker will be changed an odd number of times, it must be open at the end.

With only 20 lockers, students can simply observe this fact about the number of factors to be true for the numbers 1 to 20, which gives practice with finding factors of whole numbers. The explanation for why this is always true goes a bit beyond 4th grade.

2. The lockers that were touched by only two students are the prime numbers (since prime numbers only have two factors.) In our case they are 2, 3, 5, 7, 11, 13, 17, and 19.

3. The lockers which are touched by only three students are the squares of prime numbers (because these are the only numbers which have three factors.) In our case they are 4 and 9. (For example, 4 has three factors: 1, 2, and 4)

4. The lockers which are touched the most are those lockers whose number has the most factors. In our case 12, 18, and 20 all have 6 factors, so they were each touched 6 times. $$ 12 = 1\times 12 = 2\times 6 = 3\times 4$$ $$ 18 = 1\times 18 = 2\times 9 = 3\times 6$$ $$ 20 = 1\times 20 = 2\times 10 = 4\times 5$$

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Alignments to Content Standards: 4.OA.B.4 The 20 students in Mr. Wolf's 4th grade class are playing a game in a hallway that is lined with 20 lockers in a row. The first student starts with the first locker and goes down the hallway and opens all the lockers.

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One assessment for identifying factors and multiples that would meet UDL guidelines would be asking students to provide the factors and multiples of various numbers but at the same time, provide a number grid. A number grid can "optimize access to tools and assistive technologies" (CAST, 2010) because students can keep track of their multiples

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One standard that is commonly misunderstood in grade 3 is "CCSS.MATH.CONTENT.3.OA.B.5
Apply properties of operations as strategies to multiply and divide.² Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)" (CCSS, 2010) Most students get the commutative property but the other multiplication properties often confuse both students and their teachers alike. An online resource that could really help teachers with this content standard is this Khan Academy video with practice questions. It correctly explains the property and would be acceptable according to Van de Walle because it the target of the creator is clear. He wants to educate students for free and it explain it in a way that makes sense. (Van de Walle, )

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Sal uses the distributive property to break up 4x7 into smaller numbers.

Want to join the conversation?

What is the easiest way to learn my 7 times table? This is very hard.

8 comments Comment on elijahc457's post "What is the easiest way t..."

Many kids have difficulty with 7's, but this does not have to be you if you use a strategy.

There are multiple strategies. Here are three of them. These strategies come from the idea that multiplication is the same as counting groups of objects (distributive property).

1) If you're good with 2's and 5's, a possible way to multiply a number by 7 is to multiply it by 2, multiply it by 5, then add the results. This is based on the fact that 2 + 5 = 7.

Example: let's do 8 x 7.
8 x 2 = 16
8 x 5 = 40
16 + 40 = 56, the answer.

2) If you're good with 10's and 3's, a possible way to multiply a number by 7 is to multiply it by 10, multiply it by 3, then subtract the results. This is based on the fact that 10 - 3 = 7.

Example: let's do 8 x 7 again.
8 x 10 = 80
8 x 3 = 24
80 - 24 = 56, the answer.

3) lf you're good with 6's, a possible way to multiply a number by 7 is to multiply it by 6 and add the original number. This is based on the fact that 6 + 1 = 7.

Example: let's do 8 x 7 again.
8 x 6 = 48
original number = 8
48 + 8 = 56, the answer.

These are not the only strategies. From looking at the ideas in the three strategies l wrote, you might even come up with your own strategy that you like better.

Have a blessed, wonderful day!

4 comments Comment on Ian Pulizzotto's post "Many kids have difficulty..."

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I'm learning times table, I need to memorize them fast, how do I memorize them fast?

how do we do distributive property in multiplying, i don't understand?? Such a 7x4

1 comment Comment on saanvika's post "how do we do distributive..."

Using the distributive property can help you figure out multiplication facts that you forget.

For example, if you remember your 2's and 5's but forget your 7's, you can use this property to help you figure out your 7's (because 2 + 5 = 7).

For example:
7x4 = (5+2)x4 = (5x4)+(2x4) = 20+8 = 28.
The idea is that 7 fours equals 5 fours plus 2 fours.

Have a blessed, wonderful day!

1 comment Comment on Ian Pulizzotto's post "Using the distributive pr..."

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How can I learn my 9 times table

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well meow say you could do khan academy from kindergarten to 1 grade ahead of you

Answer Button navigates to signup page* 3 comments Comment on goldengirlsfan13's post "well meow say you could d..."

Is there an easier way to break down the problem than how he did in the video? Because even though it's easier to understand at the end, it's hard to do in my head and on paper.

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Video transcript

- [Instructor] What we're going to do in this video is dig a little bit deeper into our understanding of multiplication. And just as an example, we're going to use four times seven. And some of you might know what four times seven is, but even in this case, I think you might get something from this video because we're gonna think about how you can break down a multiplication question into simpler parts, and that's going to be useful well beyond four times seven. It's going to be useful in your future when you're tackling more and more complicated things. Now there's a couple of ways that we can visualize four times seven. My favorite way is to visualize it with angry cats, so let's bring on the angry cats. (imitates cats meowing) Yep, they're still angry. And we can see that this is a representation of four times seven. We have four rows right over here. Four rows, and each of those rows have seven cats. And so you can see that right over here, each of those rows has seven cats. Some people would call this a four by seven grid or four by seven array, however you want to view it, but if someone were to ask you what's the total number of cats, it would be four rows times seven columns, four times seven. Now another way to represent four times seven is also with a tape diagram. You might see something like this, where here, we're visualizing it as seven fours, or you could view it as four plus four plus four plus four plus four plus four plus four. Now that's all well and good and you can add that up, if you like, but what I promised you is that we would figure out ways to break down things that might simplify things in the future. Well what if you didn't know what four times seven is, but you knew what four times five is and you knew what four times two is? Well what's interesting is that seven is five plus two, so what if we tried to first figure out this many cats, so four rows and five columns right over there, and then we tried to figure out this many cats? Four rows and two columns. And you can see that it's the exact same number of cats. So one way to think about it is four times seven is the exact same thing as four times, and I'm gonna use parentheses, and that just means to do that part first, is equal to four times, instead of seven, I could write that as five plus two, 'cause that's what seven is, so all this is saying is four time seven is the same thing as four times five plus two, where you do the five plus two first, 'cause we have those parentheses around it, and five plus two is, indeed, equal to seven, and we can see that that is equivalent to the total number of cats that we have here, which we could view as what we just circled off in this orangeish pink color which would be four rows of five, so that would be equal to four times five. Four times five, and then to that, we can add this second group of cat heads or angry cat heads, and that is four rows of two. So that's four times two, and we could put parentheses if we want, just to make it a little bit more readable. Now why did I do that? Well, some folks might find four times five a little bit more straightforward. I could skip count by five. I can go five, 10, 15, 20. Also four times two might be a little bit more straightforward, and so it could be easier to say hey, this is just going to be four times five, which is 20, plus four times two, which is equal to eight, and so that is just going to be equal to 28. And you could have thought about it the same way down here with what is sometimes called a tape diagram. We could say, all right, if I have five fours, that's this mount right over here. That is the four times the five, and then I could add that to the two fours, the four times the two right over here, and that's another way to get to four times seven. So the big picture here is even if you're not dealing with four times seven, if you're not dealing with angry cats, and in most of our lives, we actually try to avoid angry cats, there might be a way to break down the numbers that you're multiplying into ones that you might be more familiar with. I'll give you one more example. Let's say someone were to ask you, well, what is six times nine? Pause this video and see if you can break this down in some useful way. Well, maybe you know what six times 10 is, and you also know what six times one is, so you could rerwrite nine as 10 minus one. Well then this would mean that six times nine is the same thing as six times 10 minus one. Based on exactly what we just did up here, that says that this whole thing is going to be the same thing as six times 10. Six times 10 minus six times one. One way to think about it is, I just distributed the six. That's the distributive property right over there, and then six times 10 is equal to 60, and then six times one is equal to six. And it might be easier for me to say, hey, 16 minus six in my head, that's equal to 54. So I know what some of you are thinking. Six times nine seems so clean, and now I've involved all of this other symbolism, symbols, and I've written down more numbers, but at the end of the day, I'm trying to give you skills for breaking down problems and including ways that you might want to do it in your head. If you're like, hey, I'm kind of foggy on what six times nine is, but six times 10, hey, I know that's 60, and six times one, of course, that's six, well what if I view this as six times 10 minus one? And then I could tackle it and get 54. And once again, you might know six times nine. You might know four times seven, but in the future, it might be useful for bigger and bigger numbers to think about how could I break this down?

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Distributive property when multiplying (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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An appropriate assessment that would pair with testing the knowledge of the multiplication properties would be if students could represent or "Illustrate through multiple media" the different properties. Students could use online tiles to show the distributive property as well as physical manipulatives. If students could do this rather than just solve a problem correctly, it would show that they really understood the property.

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An assessment that could match a this objective and would fall in line with UDL guidelines is a math running record. The running records "support planning and strategy development" because the assess the facts that the students don't know and the students explain their strategy for solving each one.

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Skip to content

Assess is rooted in the word "to sit beside." With Math Running Records you sit beside the student and find out what they know so that you can take them where they need to go! Welcome To Know More Contact Us Welcome To The Site: Math Running Records

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Math Running Records is an asset-based assessment. A Math Running Record helps teachers determine what students can do! Math Running Records is a way to assess overall basic math fact fluency. Known as the GPS of Math Fluency, it is a 3-step interview assessment that looks at accuracy, flexibility, efficiency and automaticity.

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In my experience this year, the second grade standard that seems to give students the most trouble is "CCSS.MATH.CONTENT.2.OA.B.2
Fluently add and subtract within 20 using mental strategies." (CCSS, 2010) Students that struggle with math at this level know their combinations of ten but have difficulty extending that into the teens. For example they could do 6 +4 = 10 but 16+4 stumps them and the connection is not made. A task by Illustrative Math that could help with their understanding of this skill is called Building Towards Fluency. It goes over analyzing strategies used for common addition facts. Van de Walle would approve of this online resource because "the site is associated with a reputable educational institution or organization" (Van de Walle, )

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Alignments to Content Standards: 2.OA.B.2

Task

Materials

Whiteboard or chart paper and markers

Empty number line or magnetic cubes lined up on the whiteboard, alternating colors every 5 (see solution)

List of expressions ready to write up on the board:

Actions

Write the expression on the board or chart paper. Start with 4+10.

Ask students to describe their strategy for solving the problem.

Choose one or more students to explain their strategy to the class. Represent each strategy on the board using the number line or magnetic cubes (see solution).

Once the student's strategy is understood by the class, continue with the next sum.

IM Commentary

The purpose of this task is to promote certain addition strategies that will help students learn to fluently add and subtract within 20. "Computational fluency refers to having efficient, accurate, generalizable methods (algorithms) for computing numbers that are based on well-understood properties and number relationships" (NCTM, 2000). Therefore, the focus in developing fluency should be more than the internalization of facts but on supporting students natural development of number sense so that they are able to solve computations flexibly and efficiently using their understanding of place value and relationships between numbers.

The pairs of numbers students are asked to add are carefully chosen to help students develop efficient and flexible computational skills and to develop a variety of mental math strategies for computational fluency. The first problem is a scaffold to the second. Assuming that children know the pattern of adding 10, this set encourages them to build from this understanding. The second expression has a value just 2 more than the first, and students may see the connection between the first and second problem and adding the two units. The third expression has a value 10 greater than the second and the initial representation of tens are used - but only if students offer this strategy. The next two problems are paired, since 29 is 9 more than 20. This problem also provides the opportunity to think of 29 as 30-1.

Using an open number line or a number line created by a train of cubes (see solution section below for example) will help students visualize the relationship between numbers. Pure verbal explanations can sometimes be difficult for children (and even adults) to understand. Having a visual representation allows more students to understand, take part in the discussion, and make use of the representation for further problems.

Children's natural development of numbers progress from the concrete to the abstract, from counting all (e.g. physically making four counters and then making twelve and counting all the counters to get sixteen), to counting on (e.g., counting four more starting at twelve to get to sixteen), to using part-whole (e.g. splitting apart the twelve to ten and two, and adding the two to four, then adding the ten) and relational thinking (knowing that 4 + 10 is 14 so 4 + 12 would be just two more).

This task is based on Cathy Fosnot's work using number strings - a string of related computational problems that are strategically chosen to connect specific concepts of numeracy. For more information and video demonstrations on addition and subtraction strings: http://mathcoachondemand.blogspot.com/2011/03/mental-math-strings.html http://erlc.ca/resources/resources/math_vodcasts/number_strings_to_encourage_addition_strategies_part2.php http://www.contextsforlearning.com/gradesK_3/unitBooksAnnotated.asp

Reference

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Solution

4+22 is 26. 2 and 4 is 6 and 10+ 10 + 6.

We can also use the last problem. Just add on 10 more to 4+12: so 16, then 26.Here is how it would look using the magnetic cubes:

Here is how it would look on an empty number line:

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Alignments to Content Standards: 2.OA.B.2 Materials Whiteboard or chart paper and markers Empty number line or magnetic cubes lined up on the whiteboard, alternating colors every 5 (see solution) List of expressions ready to write up on the board: Actions Write the expression on the board or chart paper.

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A first grade standard that I find to be easy to misconceive is "CCSS.MATH.CONTENT.1.OA.D.7
Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2." (CCSS, 2010) This was extremely difficult to work with my first graders on this because they treat the equal sign as an automatic answer and often put the value of the sum where an addend should go. An online activity that could work towards helping clear up the students struggling with this idea is Addition Scale by Toy Theater. This applet would be approved by Van De Walle because it is accurate or "free from errors".  (Van de Walle, )

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This online addition game helps students build fluency in addition and develop algebraic thinking skills. Try it, it’s free!

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An assessment that would pair nicely with this standard would be having students write their own true and false statements where the = sign serves as a balance. This would fit with the UDL guidelines because the assessment would be centered around clarifying "vocabulary and symbols" (CAST, 2011)

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A UDL assessment that could accompany the kindergarten standard would be asking the student to make different combinations of ten while using a manipulative like a rekenrek, a ten frame, or counters. According to UDL guidelines assessments should "Optimize access to tools and assistive technologies." (CAST,2011)

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5) Universal Design for Learning (UDL): UMTSS - Mathematics

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CCSS Writing Team. (2010). Operations & Algebraic Thinking. Operations & Algebraic Thinking | Common Core State Standards Initiative. http://www.corestandards.org/Math/Content/OA.

Universal Design for Learning (UDL): UMTSS - Mathematics. (2011). https://uen.instructure.com/courses/314069/pages/5-universal-design-for-learning-udl.

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The standard I am choosing to address for Kindergarten is: "CCSS.MATH.CONTENT.K.OA.A.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation." (CCSS,2010) As a math specialist, I have realized kids at this age have a difficult time finding and memorizing combinations of ten. To master this skill, students need to have solid number naming recognition and recognize patterns which are both difficult tasks. One online applet that I would use to tackle this objective would be a number rack linked below. If you click the minus sign on the bottom you can have a rack with 5 red beads and 5 white beads. Students can use the manipulative to find combinations of 10 by sliding the beads to opposite sides on the rack. According to Van de Walle, this internet resource is relevant because it "adds depth to your information." (Van de Walle, )

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Number Rack facilitates the natural development of children’s number sense. Rows of moveable, colored beads encourage learners to think in groups of fives and tens, helping them to explore and discover a variety of addition and subtraction strategies.