Ratio Table Strategies Lesson Plan

Warm-up

Start students off by writing a ratio based on these images of flowers and stars. The idea behind the warm up is to have students count that there are `6` flower and `9` stars, which would lead to a ratio of `6:9`. However, once simplified the ratio becomes `2:3`.

Equivalent ratios

The images are purposely placed so we can discuss equivalent ratios. On the next slide circle the first group of `2` flowers and `3` stars and write the simplified ratio of `2/3`. Then erase that circle so you can circle a group of `4` flowers and `6` stars and write the ratio `4/6`. Lastly, erase that circle and circle all the flowers and stars and write the ratio of `6/9`. The slide will now display the `3` ratios.

Ask students if these ratios are equivalent and why. Students might explain things in different ways, such as:

• Each ratio simplifies to `2/3`.
• Every time you circled more flowers and stars, you added `2` more flowers and `3` more stars, so it stayed equivalent.
• If we divide all the images into `3` equal groups, they would each have `2` flowers and `3` stars.

Student notesheet

Give each student a copy of the student notesheet. These contain the same examples as the slideshow, but it will be helpful for students to be able to draw arrows and fill in missing values as you go through the examples as a class.

Strategy `1`: Column multipliers

Show students the first example and tell them we’re looking for the column multiplier. Ask, “what can we do with `3` to get to `9`?”. Be careful for if any students respond that we can add `6`. Tell them that the rule for going from left to right needs to be the same for all rows. If we do `20 + 3`, we get `23`. But `3:9` and `20:3` are not equivalent ratios. Ultimately, we want students to recognize that we are multiplying by `3`. 

Using the “rule”

Explain to students that we can use the “rule” of multiplying by `3` from left to right to find the missing values in the second row and the fourth row. Students should multiply `20` by `3` and `50` by `3` and fill in those missing values.

“Rule” in reverse

We still have two other missing values, but they’re on the left side of the table. Go to the next slide and ask students, “if we multiplied by `3` to go from `3` to `9`, what do we need to do to go from `9` to `3`?” The key is for students to recognize that now we need to use the inverse operation and divide by `3`. Let students divide `66` and `240` by `3` to fill in the last two missing values.

Strategy `2`: Row multipliers

The next strategy deals with rows but works a little differently than the column “rule”. Show student the next example and ask them how we can go from `12` to `3`. Students should recognize that we can divide by `4`. Explain that what we do to one part of the ratio, we have to also do to the other part of the ratio, so we can divide `16` by `4` to get `4`. Now we have two completed ratios that we can work from!

You might ask the students if we can do `12-9` to get to `3`. Some students might point out that `16-9=7`. But `12:16` is not the same ratio as `3:7`.

For the next missing value, let’s start from `4` and see how we can get to `100`. Students can see what we’ll multiply `4` by `25` to get to `100`. So we need to multiply `3` by `25` to get `75`.

At this point, students have three completed ratios. Make a point to tell students that they can use any of the ratios to find new equivalent ratios, but that some make it easier than others. You might want to ask students what number on the right side of the completed ratios makes it easiest to get to `200`. Hopefully students will answer that from `100` to `200`, all you have to do is multiply by `2`!

For the last missing value we have `6` in the left column. Allow students to pick whichever ratio they want to use to find the last missing value, `8`!

Strategy `3`: Adding rows

For the last strategy we have a new table. Consider going back to the warm up and reminding students that as we added equivalent ratios, we got another equivalent ratio! Like how `2` flowers and `3` stars, combined with `4` flowers and `6` stars, gave us `6` flowers and `9` stars. This can show students how all the ratios are equivalent. 

Then ask if they think they can use a similar strategy here. If students need more support getting started, highlight the second and third row and ask how we can use those ratios to find the missing value in the fourth row. We want students to recognize that since `10 + 50 = 60`, we can add `6` and `30` to fill in the missing value of `36`. 

Once students see how this is done, they tend to take off! Allow students to work together to find the rest of the missing values by adding different rows.