Tag Archives: proportions

CCSS MATH Coherence Map – 3rd Grade Groups of Objects to HS Slope

A hidden treasure is the Common Core of State Standards Math Coherence Map. It is an interactive flow chart that shows connections between the various standards.

  • If you are teaching math, you can see the connections between what you are teaching, what was taught previously, and how you are preparing students for their future math education.
  • If you are a special education teacher, you can see the progression of prerequisite skills. If you write IEP objectives for grade level standards, you can address the prerequisite standards and progress made through these prerequisites can show progress towards mastery of the IEP objectives.

In this post, I show the progression from 1st grade standard on the = sign and 2nd grade standard of repeated addition all the way to interpreting slope in high school math.

After clicking “Get Started” you will narrow your focus to the grade, the cluster, and then the math domain.

The flow chart shows connections between a selected standard and others, including prerequisites. In this case 8.F.B.4 – 8th grade content that is a prerequisite for the high school math work. Click on the 8.F.B.4 standard and it pops up (below right).

In turn, the 8th grade standard is connected to a 7th grade prerequisite regarding ratios and proportions.

Notice that the 7th grade standard, 7.RP.A.2, appears to be a dead end (bottom left). I picked up the path by clicking on Grade seen at the top left of the screenshot and made my back to that standard and the connections to prerequisites appeared. (Same happened in 3rd grade shown further down in this post.)

The path continues from ratio and proportions in 7th grade to unit rate in 6th grade, multiplication word problems and multiplication in elementary school.

I want to emphasize that students are working on unit rate and slope problems in ELEMENTARY SCHOOL! 3.OA.A.1 below addresses groups of objects model for multiplication and 4.OA.A.2 addresses word problems involving multiplication.

I was recently working with a student entering middle school on multiplication word problems. To unpack the word problems and the concept of multiplication in context, I had her draw (photo below) groups and groups of objects to help her identify the unit rate (although we don’t use that term yet). This work will impact her math education through the high school math and even into college (slope has been a common gap for the college students in the math courses I taught at various colleges and universities).

This approach I used with the student could be used for high school students, especially those with special needs.

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K-12 Math Education is a Line of Dominoes

Critical dominoes in math education start falling in 6th and 7th grade with the last ones falling in college. If you have a student who struggles with math and is entering or returning to middle school, now is the time to intervene to avoid more serious issues related to math education in the future. If your student is not going to college or is not accessing the general curriculum, I suggest you read this.)

Below is a chart showing the different categories of Common Core of State Standards (CCSS) math (called domains) at different grade levels. For the majority of students who will attend college, the traditional algebra based sequence (algebra 1, algebra 2, and maybe pre-calculus, calculus) is the path of math courses to be taken. Given this, for students who struggle in math but have a post-secondary education as a goal, the domains I emphasize in middle school are Expressions and Equations, Ratios and Proportional Relationships, and Functions. For high school, I emphasize Algebra and Functions.

Looking at the overviews for CCSS math standards (below) you can see the dominoes line up.

  • In 6th grade, Ratios and Proportions are an entry point for Functions in 8th grade which leads to Functions in high school.
  • In 6th grade, Expressions and Equations are the entry point for Expressions and Equations in 7th and 8th grade, which lead to Algebra in high school.

If your student is struggling with the middle school topics I cited and the gaps are not filled, the struggle will be carried with them into high school and into college.

I recommend the following:

  • Focus IEP math objectives on the priority units of the math curriculum, as cited above.
  • Ask for examples of mastery for the objectives to help you evaluate progress and mastery. Have this in place from day 1.
  • Focus on study skills, not just content mastery.

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Making Proportions Meaningful (and Therefore Accessible)

A student reported to our schools math lab where I reside. He had a handout on proportions shown in the photo below and stated that he didn’t know what to do.

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I find that in the vast majority of situations like this the student lacks the conceptual understanding of the topic. As is typically the case, I started my sessions with the student by focusing on something he more intuitively understood. Teens know money, phones, games, music and food.

In this case I started by showing him a photo on my phone shrunk then enlarged the photo and talked about how I could double the size of the photo. We talk about what doubling means then I show him a handout with the photo in two sizes (below).

I explained that the small photo was 3×2 inches and that I wanted to enlarge it. The bottom of the big photo is 6″ but I needed to figure out the height (vertical length) which is marked with an X.

I had him figure out the height (4). Then I explained that proportional means the shape is the same but bigger or smaller. In this case both the side and bottom were multiplied by 2. Then I showed him the “mathy” way of doing the problems. This progressed towards the handout he brought into math lab. By the end he was doing the proportions independently.

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Analogies: Making Math Meaningful

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Math is an esoteric subject for most people. Good instruction makes information meaningful. One method for making information meaningful is to connect new information to prior experience.

In this situation the new information involves determining whether shapes are similar (see photo below). One example of student prior experience with this topic would be shrinking people down. In the photo above I use Mini Me and Dr. Evil and their respective (and fabricated) weights and shoe sizes as measures that will eventually give way to measures of sides of a polygon (below). When working on the problem below the students can be prompted by recalling the analogy of Mini Me and Dr. Evil.

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Dragon Box for Equations and Proportions

My 5 year old son using Dragon Box to solve a proportion

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