This post provides details about a handout for simplifying rational monomial expressions. It incorporates a couple strategies to make the simplification of rational monomial expressions more accessible. The strategies include address prerequisites skills ahead of time, chunking, and scaffolding. This incrementally walks the students through the steps.
The Pages of the Handout
The handout has 3 pages.
Page 1 is an initiation with two parts. There is a review of prerequisite skills aligned with the new topic. There is also a preview of the new topic with scaffolding to separate the factors into individual fractions.
Page 2 provides a Before and Now to draw upon student prior knowledge of simplifying using exponents rules. This is followed by scaffolded steps to separate the expression into individual fractions for each type of term (e.g., Xs). This provides a load reduction for what the student has to focus on.
Page 3 involves negative and 0 exponents with an additional step to address each.
Functions are perhaps the most prevalent and important topic covered in secondary math, aside from maybe 1 variable linear equations. The concept of a mathematical function is challenging for many students. This post provides details about a meaning making approach to introducing functions.
The introduction is presented on a Google Jamboard, to allow for movement in the pairing of inputs and outputs. It starts with analogies pairing of items using a gumball machine and a Coke machine and proceeds incrementally towards the various representations. The functions are contrasted with examples of relationships that are not functions.
Slides of the Jamboard
Slides1 and 2 present the gumball and Coke machines. Students can move the items to see how a quarter can result in 2 different color gumballs while the Coke button results in only 1 output.
In slides 3 and 4, the use of an hourly wage introduces input and output with quantities. Slide 4 shows two different pay amounts for the same number of hours worked. This taps into prior knowledge.
The sequencing progresses through
Each includes an example and a non-example.
The last slide provides a sorting activity.
Access to Jamboard
Here is the link. To access the Jamboard, you need to make a copy.
The effort to provide intervention to fill in gaps is challenging for different reasons. One reason is the effort to balance support for current content while filling in gaps. This post shows an example of how to fill in gaps while working through the current topic.
Various rubrics used to assess teacher instruction includes an effort to build on or connect to prior knowledge. If the student has gaps with prior knowledge, the lesson becomes less accessible for students with the gaps. Previously, I addressed how to support both current content and fill in gaps. The idea is to systematically fill in gaps by addressing prerequisite skills as they arise in new lessons.
The handout out below shows an example of how this can play out. The first page is used as a do now for the content presented on page 2. If you are teaching a student how to solve 1 step equations and are moving into integers, page 1 is a a means of supporting the new content while filling in possible gaps. The first image shows the student will need to evaluate -13 – 3 as part of the solving in the lesson. This can be addressed in the do now, as shown in the 2nd image, page on the right. (Notice all the problems on page 1 are steps to solve on page 2 problems.) This is useful for students with special needs and for differentiation.
Fractions are challenging. Multiplying fractions is really challenging! This post presents a Google Jamboard to introduce students to the concept of multiplication of fractions.
The artifact is chunked to incrementally move from multiplication of whole numbers to whole number and fraction to multiplication of fractions. The representation of multiplication as number of objects in a group times number of groups is the structure used throughout. Cookies on a plate is the context used to draw upon prior knowledge and make the idea more concrete.
This serves as an introduction. Each chunk can be followed by practice before moving on to the subsequent chunk.
The Jamboard starts with a representation of multiplication as groups of objects, first with the number of objects in a group and the number of groups. This is presented first as cookies per person to connect to prior knowledge. Then presented per plate as the plate is subsequently used to model the fractions.
First, whole number times a fraction is presented. This allows for a connection to prior knowledge and introduces fractions in this representation. There are still 6 cookies per group, but now there is only 1/2 a group.
The students can move the cookies onto the plate to see the group of objects. Then they can cut the group in half.
To help make sense of the fractions used in the multiplication of two fractions, the fractional parts of the cookies are presented first.
For multiplication of fractions, the process is the same. There is 1/4 of a cookie in each group, then there is 1/2 a group. As was done previously, 1/2 the group is removed. Conceptually, you can explain to the students that they have 1/4 of a cookie and they split it with a friend.
I work with students from elementary school to college. At all levels there are many students who struggle with multiplication. When I work with a student on multiplication I focus on skip counting as opposed to multiplication facts. Skip counting connects to multiplication as repeated addition, which is the foundation for scale and proportion. To do this, I scaffolded the skip counting and connect it to prior knowledge. This post provides details for this approach, which is presented in a handout.
The first page of the handout provides an overview.
Sections of the Handout
Note: the image of the 10s shows mistakes that are not in the actual handout.
Access to the Handout
Here is a link to the document. As stated in the overview on page 1, the elements of this handout may be cropped out and used individually. At least, the scaffolded rows can be covered to prevent the students from copy from previous rows completed.
I am interested in feedback on how to make this more useful or effective.
Tenths vs Tens…Hundredths vs Hundreds. Problematic for many students. I believe this is a conceptual problem. This post provides an approach to unpack the concepts through money in a scaffolded handout.
Money is likely prior knowledge for many if not most students, and is a relevant context. This handout attempts to leverage interest or knowledge of money to unpack decimal place values. In the first page, the concept of “tenth” is addressed with dimes as 1/10 of a dollar. Similarly, “hundredth” is addressed with pennies as 1/100 of a dollar. A key point to consider is that US monetary system base unit is a dollar. More on that in the other pages.
Hundreds to Hundredths
The handout aligns each place value with the appropriate currency. This is followed by writing each number in numeric form and then word form with the place value table as a guide. To enhance the word part, you can highlight the each place value in money, digit, and word in the same color (e.g., the “2” in yellow).
Also, note the shading. The dollar as the base unit is in the center and shaded the darkest. The tens and tenths are shaded the same as they are a factor of 10 from ones. (I don’t reference the term with students.) Same for hundreds and hundredths.
My focus is on working with students with special needs. Many struggle with rote memorization, including for multiplication facts. I find that skip counting, with scaffolded support in the learning process, provides them access to multiplication and therefore division. To access division, I use an approach of skip counting to find a missing factor and then connect this to division. This post provides details of a handout using this approach.
This handout focuses on connections to prior knowledge of skip counting and finding a missing factor. The students then make an explicit connection by rewriting division problems as missing factor problems. The handout is linked at the bottom.
If students are struggling with multiplication, they are likely having trouble with skip counting. I start with a warm up on skip counting with the numbers that are easiest for students to skip count. Note: you can start with 2, 5, 10 only if necessary.
I have students solve a missing factor problem using a provided skip counting row. Then they are shown that the problem can be rewritten as a division problem which has the missing factor as the answer. That is, division is another way to write a missing factor problem. You can use factor tree handouts and have students practice rewriting the problem as a division problem. Note: I see that most worksheets are used for prime factorization. Use the first two branches as shown in the image below.
The students are then presented a math sentence only for missing factor. They are to solve for the missing factor. Then they rewrite the math sentence into a division sentence and solve again. I have a separate column to help emphasize that they are lookin to solve a division problem. They have to see the division problem in isolation and then write the quotient.
Finally, the students are presented division problems and rewrite as a missing factor problem. Their mental process can be as follows: “2 times what gives me 10?” and then they skip count by 2s until they reach 10. This can be supported with multiples rows as shown in the factor tree page. A blank page is provided. You can give students a division worksheet and have them copy the problems into the handout.
Absolute value is a challenging for many students. Absolute value equations add an extra layer of challenge. This post presents a scaffolded handout. It starts with an initiation addressing absolute value and then is followed by scaffolded steps for solving that address written and mental steps.
The scaffolding walks students through mental steps to unpack the underlying concepts. The first is a general concept of absolute value. The next two are specific to the problem. At the end, students are prompted to check the solutions to reinforce the concept of two possible values, one being negative and one positive. Note: this is not addressing the case of =0.
The scaffolded handouts are preceded by an initiation. The focus is to unpack the concepts underlying the equation steps. The key is for the students to understand that there are two possible solutions (aside from =0). The handout addresses the reason for two solutions and provides work with the absolute value symbol in this context. Here is a link to a Jamboard that helps unpack the concept of absolute value.
There are 3 practice problems with the template. I use 2 and 6 to help compare the different situations. There is another page with blank templates.
Solving division equation proportions is challenging for two reasons. First, it involves fractions. Second, for some reason, many students struggle with solving division equations and a proportion is a more complex version. This post outlines a scaffolded handout to guide students through solving by multiplying both sides.
The handout provides support in two ways. First, it draws upon prior knowledge of solving division equations Second, it scaffolds the initial multiplication. This post is in contrast to another in which I share scaffolding for cross multiplication, which is helpful if a variable term is a binomial.
Page 1 focuses on solving division equations to draw upon prior knowledge and to introduce the scaffolding. You can have students write a 1 under the factor.
This page uses the same scaffolding, but now with proportions. The focus is on multiplying the numerator, as was done on the first page.
Word problems are challenging. In middle school and high school, word problems for proportional relationships and linear functions are particularly challenging. This post shows an visualization approach to unpacking the unit rate component of such word problems.
Visualizing Unit Rate
Below is an excerpt from a handout used to introduce such word problems. It starts with the unit rate and unit quantity provided. This is more accessible for students and allows for a visual representation. This allows students to “see” the multiplication and make a connection to the word problem.
This is followed by situations in which the unit quantity is unknown. Students still draw a visual, but use the 3 dot symbol for etc. to indicate the unknown quantity.
The same problems are used but students now write the symbolic representation. First they write the expression with both factors provided, then with the unknown. The “?” emphasizes there is an unknown quantity before writing the variable.