SWBAT use table and diagrams to make equivalent ratios.

Making equivalent ratios using multiple representations.

10 minutes

The students will be using their memory box to activate prior knowledge about ratio tables. Pose the question: In a moment I am going to have you list everything you remember about ratio tables. I’m going to give you 10 seconds of think time. (remember, thinking happens in our heads, not out our mouths : ) Once the think time is over, give the students a couple of minutes to independently write down everything they remember about ratio tables. When the time is up, allow students to look in their tool box under ratio tables. Give them another minute to look it over, but not write anything down. Then, give them another minute, with notes closed, to write down a few more things they can remember.

When students are done, have them complete an “I have that”. One student will read something from their memory box. If the other students have that they say out loud “I have that” and check it off their list. If they don’t have it, then they add it to their list. This continues until all items are marked off the list.

15 minutes

Indeed, all fractions are ratios. Fractions are considered rational numbers which has the base word ratio in it. At this time, it would be interesting to see if students could come up with a counter example.** (SMP 3)** Allow students to debate this with a partner and ask for responses. Reminder: students must use mathematical reasoning and language when arguing their point. Ratios, in common core for 6th grade is a major shift, and our focus should be on ratios as a comparison of two quantities through division and not ratios as fractions. We will make this shift by using tables and diagrams to make our equivalent ratios.

Begin by asking students the following question. If the ratio of females to males at the University of Illinois campus is 48:52 can you tell what it means? To make this ratio easier to understand, find an equivalent ratio by expressing this in simplest terms. At this time, in this situation, would it be helpful to use a table? And why is it helpful **(SMP 4 and 5)** Help students set up the table, remind students that when we want to express a ratio in simplest terms, we can divide to simplify. We stop dividing when the only common factor is 1. Allow students time to put the information in a ratio table. Once they have simplified the ratio to 12 out of 13, have them explain the meaning of the ratio. For every 12 females on campus there are 13 males which means that there is approximately the same amount of females as males.

We want to develop the multiplicative relationship which means the relationship is one in which the number is multiplied to get the second number. In addition, the same multiplicative relationship can be applied to other situations. Using the ratio table helps develop this multiplicative relationship. In general, both numbers in the table need to be multiplied or divided by the same number in order to maintain equivalence.

20 minutes

For the next 3 slides, I’m going to level the problems so that they go from easy to more challenging. Before starting each ratio, have the students determine the ratio first.

Slide # 4 : This table will begin with a simpler ratio and no “skipped” values in the left-hand column. When students are using the ratio table have them write out the steps to get to the missing quantity.

Slide #5 : This slide is presented with skipped quantities on the left-hand column. Students will need to use their ratio table strategies to find the missing values. Have the students write out their steps to finding these missing quantities.

Slide #6 : This is the most challenging table. The values do not go in any order. Students will have to have a good understanding of ratio table strategies to find the missing values. Have the students come up with a title and labels for this table too.

Again, have the students write down the steps to solve. Students may want to create their own ratio table using different strategies to solve and that is fine.

25 minutes

Using the** Numbered Heads Together** power point, have students first solve the problems independently and then check with their teammates.**(SMP 3)** Call students from the group, at random, to show the answer.

15 minutes

Have students complete a Comprehension Menu for Equivalent Ratios. The students will be responding to 4 learning style type questions. Have students examine each question (or prompt) to determine the kinds of thinking that responding to the question will require. Give students time to respond and then reflect on their comfort with thinking and working in each style. Have students put an X in the box that was the most comfortable for them to answer. **(SMP1,2,3,5)**